PLATE  L 


COMPARATIVE  MAGNITUDES  OF  THE  PLANETS, 

1,  MERCURY.  5.  URANUS. 

2,  MARS.  6,  NEPTUNE. 

3,  VENUS  7,  SATURN. 

4,  EARTH  8.  JUPITER. 


THE 


ELEMENTS 


THEORETICAL  AND  DESCRIPTIVE 


ASTRONOMY, 


the  tt 


BY 

CHARLES  J.  WHITE,  A.M., 
/ 1 

ASSISTANT   PROFESSOR  OF   MATHEMATICS   IN   HARVARD   COLLEQB 


FOURTH  EDITION,  REVISED. 


PHILADELPHIA: 

CLAXTON,  EEMSEN  &  HAFFELFINGER. 

1880. 


NA/S 


Entered  according  to  Act  of  Congress,  in  the  year  1869,  by 
CHARLES  J.  WHITE, 

ir>  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  District,  of 
Maryland. 


PREFACE  TO  THE  THIRD  EDITION. 


first  edition  of  this  work  was  published  in  1869,  to  meet 
the  requirements  of  the  students  of  the  United  States  Naval 
Academy.  In  preparing  it,  I  endeavored  to  present  the  main 
facts  and  principles  of  Astronomy  in  a  form  adapted  to  the 
elementary  course  of  instruction  in  that  science  which  is  com- 
monly given  at  colleges  and  the  higher  grades  of  academies.  I 
selected  those  topics  which  seemed  to  me  to  be  the  most  important 
and  the  most  interesting,  and  arranged  them  in  the  order  which 
experience  had  led  me  to  believe  to  be  the  best.  The  third  edi- 
tion of  the  book  is  issued  with  no  change  in  the  general  plan, 
and  with  only  those  changes  which  the  advance  of  astronomical 
knowledge  in  the  last  six  years  renders  necessary. 

In  the  descriptive  portions  of  the  work,  I  have  endeavored  to 
give  the  latest  information  upon  every  topic  which  is  introduced. 
On  not  a  few  points  the  opinions  of  competent  observers  are  by 
no  means  the  same ;  and  on  these  points  I  have  endeavored  to 
give,  as  far  as  possible,  the  various  opinions  which  now  exist. 
The  distances  and  the  dimensions  of  the  heavenly  bodies  are 
given  to  correspond  with  the  value  of  the  solar  parallax  which 
is  at  present  adopted  in  the  American  Ephemeris ;  the  recent 
theories  upon  the  connection  of  comets  and  meteors,  the  principles 
of  spectroscopic  observation,  and  the  conclusions  concerning  the 


IV  PREFACE. 

• 

constitution  and  the  movements  of  the  heavenly  bodies  which 
such  observation  induces,  are  given,  it  is  hoped,  in  sufficient 
detail. 

No  clear  conception  of  the  processes  Dy  which  most  of  the 
fundamental  truths  of  Astronomy  have  been  established  can  be 
attained  without  some  knowledge  of  Mathematics.  I  have  en- 
deavored, however,  to  confine  the  theoretic  discussions  within  the 
limits  of  such  moderate  mathematical  knowledge  as  may  fairly 
be  expected  in  those  readers  for  whom  the  treatise  is  intended. 
Certain  definitions  and  formulae,  with  which  the  student  may 
possibly  not  be  familiar,  will  be  found  in  the  Appendix ;  and, 
with  this  aid,  I  believe  that  every  portion  of  the  work  can  be 
read  without  difficulty. 

In  the  preparation  of  the  work,  many  authorities  have  been 
consulted ;  the  principal  ones  being  Chauvenet's  Manual  of 
Spherical  and  Practical  Astronomy,  and  Chambers's  Descriptive 
Astronomy.  The  treatise  has  been  used  as  a  text-book  in  the 
United  States  Naval  Academy,  the  Massachusetts  Institute  of 
Technology,  Harvard  College,  and  other  institutions  ;  and  I  am 
indebted  to  officers  of  these  institutions  for  many  valuable  sug- 
gestions as  to  errors  and  improvements.  I  trust  that  this  new 
edition  will  be  found  to  be  free  from  mistakes,  and  that  it  will  be 
useful,  not  only  to  the  class  of  students  for  whom  it  is  especially 
prepared,  but  to  others  who  may  wish  to  know  the  general  prin- 
ciples and  the  present  state  of  the  science  of  Astronomy. 
Harvard  College,  Cambridge,  Mass.,  1875. 


CONTENTS. 


The  Greek  Alphabet PAGE  ix 

CHAPTER  I. 

GENERAL  PHENOMENA   OF  THE   HEAVENS.      DEFINITIONS.      THE 
CELESTIAL   SPHERE. 

The  heavenly  bodies.  Astronomy.  Form  of  the  earth.  Diurnal  mo- 
tions of  the  heavenly  bodies.  Eight,  parallel,  and  oblique  spheres. 
Definitions.  Theorems.  The  astronomical  triangle.  Spherical  co- 
ordinates. Vanishing  lines  and  circles.  Spherical  projections 11 

CHAPTER  II. 

ASTRONOMICAL  INSTRUMENTS.      ERRORS. 

The  clock :  its  error  and  rate.  The  chronograph.  The  transit  instru- 
ment: its  construction,  adjustment,  and  use.  The  meridian  circle. 
The  reading  microscope.  Fixed  points.  The  mural  circle.  The 
altitude  and  azimuth  instrument.  Method  of  equal  altitudes.  The 
equatorial.  The  sextant.  The  artificial  horizon.  The  vernier. 
Other  astronomical  instruments.  Classes  of  errors 28 

CHAPTER  III. 

REFRACTION.      PARALLAX.      DIP  OF  THE  HORIZON. 

General  laws  of  refraction.  Astronomical  refraction.  Geocentric  and 
heliocentric  parallax.  The  dip  of  the  horizon 51 

CHAPTER  IV. 

THE  EARTH.      ITS  SIZE,   FORM,   AND  ROTATION. 

Measurement  of  arcs  of  the  meridian  by  triangulation.  Spheroidal 
form  of  the  earth.  Its  dimensions.  Its  volume,  density,  and 
weight.  Rotation  of  the  earth.  Change  of  weight  in  different 
latitudes.  Centrifugal  force.  The  trade-winds.  Foucault's  pendu- 
lum experiment.  Linear  velocity  of  rotation 59 


Vi  CONTENTS, 

CHAPTEK  V. 

LATITUDE  AND  LONGITUDE. 

Four  methods  of  finding  the  latitude  of  a  place.  Latitude  at  sea. 
Reduction  of  the  latitude.  Longitude.  Greenwich  time  by  chro- 
nometers, celestial  phenomena,  and  lunar  distances.  Difference  of 
longitude  by  electric  and  star  signals.  Longitude  at  sea.  Compa- 
rison of  the  local  times  of  different  meridians PAGE  72 

CHAPTEK  VI. 

TIIE  SUN.      THE   EARTH'S  ORBIT.      THE  SEASONS.      TWILIGHT.      THE 
ZODIACAL  LIGHT. 

The  eciiptic.  Distance  of  the  sun  from  the  earth  determined  by  transits 
of  Venus.  Magnitude  of  the  sun.  The  earth's  orbit  about  the  sun. 
The  seasons.  Twilight.  Rotation  of  the  sun,  and  its  constitution. 
The  zodiacal  light 83 

CHAPTER  VII. 

SIDEREAL  AND  SOLAR  TIME.      EQUATION   OF   TIME.      THE   CALENDAR. 

The  sidereal  and  the  solar  year.  Relation  of  sidereal  and  solar  time. 
The  equation  of  the  centre.  The  equation  of  time.  Astronomical 
and  civil  time.  The  calendar 100 

CHAPTER  VIII. 

UNIVERSAL   GRAVITATION.      PERTURBATIONS   IN   THE   EARTH'S   ORBIT. 
ABERRATION. 

The  law  of  universal  gravitation.  The  mass  of  the  sun.  The  earth's 
motion  at  perihelion  and  aphelion.  Kepler's  laws.  Precession. 
Nutation.  Change  in  the  obliquity  of  the  ecliptic.  Advance  of  the 
line  of  apsides.  Diurnal  and  annual  aberration.  Velocity  of  light. 
Aberration  a  proof  of  the  earth's  revolution 107 

CHAPTER  IX. 

THE  MOON. 

The  orbit  of  the  moon,  and  perturbations  in  it.  Variation  of  the 
moon's^meridian  zenith  distance.  Distance,  size,  and  mass  of  the 
moon.  Augmentation  of  the  semi-diameter.  The  phases  of  the 
moon.  Sidereal  and  synodical  periods.  Retardation  of  the  moon. 
The  harvest  moon.  Rotation ;  librations  and  other  perturbations. 
Tlu-  lunar  cycie.  General  description  of  the  moon 120 


CONTENTS.  Vll 

CHAPTEK  X. 

LUNAR  AND  SOLAR  ECLIPSES.      OCCULTATIONS. 

Lunar  eclipses.  The  earth's  shadow.  Lunar  ecliptic  limits.  Solar 
eclipses.  The  moon's  shadow.  Solar  ecliptic  limits.  Cycle  and 
number  of  eclipses.  Occultations.  Longitude  by  solar  eclipses  and 
occultations PAGE  135 

CHAPTEK  XI. 

THE  TIDES. 

Cause  of  the  tides.  Effect  of  the  moon's  change  in  declination.  General 
laws.  Influence  of  the  sun.  Priming  and  lagging  of  tides.  The 
establishment  of  a  port.  Cotidal  lines.  Height  of  tides.  Tides  in 
bays,  rivers,  &c.  Four  daily  tides.  Other  phenomena 146 

CHAPTER  XII. 

THE  PLANETS   AND   THE  PLANETOIDS.      THE  NEBULAR  HYPOTHESIS. 

Apparent  motions  of  the  planets.  Heliocentric  parallax.  Orbits  of 
the  planets.  Inferior  planets.  Direct  and  retrograde  motion.  Sta- 
tionary points.  Evening  and  morning  stars.  Elements  of  a  planet's 
orbit.  Heliocentric  longitude  of  the  node.  Inclination  of  the  orbit. 
Periodic  time.  Mercury.  Venus.  Transits  of  Venus.  Superior 
planets:  their  periodic  times  and  distances.  Mars.  The  minor 
planets.  Bode's  law.  Jupiter:  its  belts,  satellites,  and  mass.  Saturn 
and  its  rings.  Disappearance  of  the  rings.  Uranus.  Neptune. 
The  nebular  hypothesis 155 

CHAPTEE  XIII. 

COMETS  AND   METEORIC  BODIES. 

General  description  of  comets.  The  tail.  Elements  of  a  comet's  orbit. 
Number  of  comets  and  their  orbits.  Periodic  times.  Motion  in 
their  orbits.  Mass  and  density.  Periodic  comets.  Encke's  comet. 
Winnecke's  or  Pons's  comet.  Brorsen's  comet.  Biela's  comet. 
D' Arrest's  comet.  Faye's  comet.  Mechain's  comet.  Halley's 
comet.  Remarkable  comets  of  the  present  century.  The  great 
comet  of  1811.  The  great  comet  of  1843.  Donati's  comet.  The 
great  comet  of  18G1.  Meteoric  bodies.  Shooting  stars.  The  No- 
vember showers.  Height  and  orbits  of  the  meteors.  Detonating 
meteors.  Aerolites.  Connection  of  comets  and  meteoric  bodies 187 


Viii  CONTENTS. 

CHAPTEK  XIV. 

THE  FIXED  STARS.      NEBULA.      MOTION  OF  THE  SOLAR  SYSTEM.      REAL 
MOTIONS   OF  THE  STARS. 

Proper  motions  of  the  fixed  stars.  Magnitudes.  Constellations. 
Constitution  of  the  stars.  Distance  of  the  stars.  Bessel's  differen- 
tial observations.  Heal  magnitudes  of  the  stars.  Variable  and  tem- 
porary stars.  Double  and  binary  stars.  Colored  stars.  Clusters. 
Kesolvable  and  irresolvable  nebulae.  Annular,  elliptic,  spiral,  and 
planetary  nebulae.  Nebulous  stars.  Double  nebulae.  The  Ma- 
gellanic  clouds.  Variation  of  brightness  in  nebulre.  The  milky  way. 
Number  of  the  stars.  Motion  of  the  solar  system  in  space.  Real 
motions  of  stars  detected  with  the  spectroscope PAGE  213 

APPENDIX. 

Mathematical  definitions,  theorems,  and  formulae 242 

Kirkwood's  law 247 

Chronological  history  of  astronomy 248 

Sketch  of  the  history  of  navigation 255 

Table        I. — Elements  of  the  planets,  the  sun,  and  the  moon 256 

II.— The  earth... 257 

"        III.— The  moon 257 

"        IV.— Elements  of  the  satellites 258 

.  "          V. — The  minorplanets 259 

"        VI. — Schwabe's  observations  of  the  solar  spots 261 

"      VII.— Elements  of  the  periodic  comets 261 

"    VIII.— Transits  of  the  inferior  planets 262 

"        IX. — Stars  whose  parallax  has  been  determined 262 

"          X.— The  constellations 263 

"        XI.— Examples  of  variable  stars 266 

"      XII.— Examples  of  binary  stars 267 

[NDEX ..  269 


THE  GKEEK  ALPHABET. 

The  following  table  of  the  small  letters  of  this  alphabet  is 
given  for  the  use  of  those  readers  who  are  unacquainted  with 
the  Greek  language. 


a 

Alpha. 

V 

Nu. 

P 

Beta. 

5 

Xi. 

v 

Gamma. 

0 

Omicron. 

$ 

Delta. 

7t 

Pi. 

e 

Epsilon. 

P 

Rho. 

? 

Zeta. 

(7 

Sigma. 

n 

Eta. 

* 

Tau. 

$or  0 

Theta. 

U 

Upsllon. 

i 

Iota. 

<P 

Phi. 

X 

Kappa. 

z 

Chi. 

a, 

Lambda. 

$ 

Psi. 

P 

Mu. 

G) 

Omega. 

ASTRONOMY. 


CHAPTER  I. 

GENERAL   PHENOMENA   OF   THE   HEAVENS.      DEFINITIONS. 

1.  The  heavenly  bodies  are  the  sun,  the  planets,  the  satellites 
of  the  planets,  the  comets,  the  meteors,  and  the  fixed  stars. 

The  planets  revolve  about  the  sun  in  elliptical  orbits,  and  the 
satellites  revolve  in  similar  orbits  about  the  planets.  The  earth 
is  a  planet,  as  we  shall  see  hereafter,  and  the  moon  is  its  satel- 
lite. The  comets  revolve  about  the  sun  in  orbits  which  are 
either  ellipses,  parabolas,  or  hyperbolas.  Comparatively  little 
is  known  with  any  degree  of  certainty  about  the  meteors;  but 
it  is  probable  that  they  too  revolve  about  the  sun. 

The  sun,  the  planets,  the  satellites,  and  the  comets  constitute 
what  is  called  the  solar  system.  The  fixed  stars  are  bodies  which 
lie  outside  of  this  system,  and  preserve  almost  precisely  the 
same  configuration  from  year  to  year. 

The  heavenly  bodies  may  be  considered  to  be  projected  upon 
the  concave  surface  of  a  sphere  of  indefinite  radius,  the  eye  of 
the  observer  being  at  the  centre  of  the  sphere.  This  sphere  is 
called  the  celestial  sphere. 

2.  Astronomy  is   the   science  which   treats  of  the   heavenly 
bodies.     It  may  be  'divided  into  Theoretical,  Practical,  and  De- 
scriptive Astronomy. 

Theoretical  Astronomy  may  be  divided  into  Spherical  and 
Physical  Astronomy. 

Spherical  Astronomy  treats  of  the  heavenly  bodies  when  con- 
sidered to  be  projected  upon  the  surface  of  the  celestial  sphere. 
It  embraces  those  problems  which  arise  from  the  apparent  diur- 

ll 


12  FORM   OF   THE   EARTH. 


nal^inpiion  pf^fejfeay.enly  bodies,  and  also  those  which  arise 
from.  anyichaiiges  in.  the  apparent  positions  of  these  bodies  upon 


ffca  ceiestiaL&phere. 
Physical  Astronomy  treats  of  the  causes  of  the  motions  of  the 
heavenly  bodies,  and  of  the  laws  by  which  these  motions  are 
governed. 

Practical  Astronomy  treats  of  the  construction,  adjustment, 
and  use  of  astronomical  instruments. 

Descriptive  Astronomy  includes  a  general  description  of  the 
heavenly  bodies;  of  their  magnitudes,  distances,  motions,  and 
configuration  ;  of  their  appearance  and  structure  ;  of,  in  short, 
every  thing  relating  to  these  bodies  which  comes  from  observa- 
tion or  calculation. 

3.  Form  of  the  Earth.  —  We  may  assume,  at  the  outset,  that 
the  form  of  the  earth  is  very  nearly  that  of  a  sphere.  The  fol- 
lowing are  some  of  the  reasons  which  may  be  given  for  such  an 
assumption  :  — 

(1.)  If  we  stand  upon  the  sea-shore,  and  watch  a  ship  which 
is  receding  from  the  land,  we  shall  find  that  the  topmasts  remain 
in  sight  after  the  hull  has  disappeared.  If  the  surface  of  the 
sea  were  merely  an  extended  plane,  this  would  not  happen  ;  for 
the  topmasts,  being  smaller  in  dimensions  than  the  hull,  would 

in  that  case  disappear  first. 
The  supposition  that  the  sur- 
face of  the  sea  is  curved, 
however,  fully  accounts  for 
this  phenomenon,  as  may  be 
seen  in  Fig.  1.  Let  the  curve 
Fig.i.  CBG  represent  a  portion  of 

the  earth's  surface,  and  let  A 

be  the  position  of  the  observer's  eye  :  it  is  at  once  evident  that 
no  portion  of  the  ship,  S,  will  be  visible  which  is  situated  below 
the  line  AH,  drawn  from  A  tangent  to  the  earth's  surface  at  C. 
The  same  figure  also  shows  why  it  is  that  when  a  ship  is  ap- 
proaching land  any  object  on  shore  can  be  seen  from  the  top- 
masts before  it  is  seen  from  the  deck. 

(2.)  At  sea,  the  visible  horizon  everywhere  appears  to  be  a 
circle.  This  also  is  easily  explained  on  the  supposition  that  the 


DIURNAL    MOTION. 


13 


eartli  is  spherical  in  form;  for  if,  in  Fig.  1,  the  line  AH  is 
turned  about  the  point  A,  and  is  continually  tangent  to  the 
sphere,  the  points  of  tangency,  C,  D,  E,  &c.,  will  form  the  visible 
horizon  of  the  spectator,  and  will  evidently  constitute  a  circle. 

(3.)  A  lunar  eclipse  occurs  when  the  earth  is  situated  between 
the  moon  and  the  sun.  Now  the  shadow  which  the  earth  at 
such  a  time  casts  upon  the  moon  is  invariably  circular  in  form : 
and  a  body  which  in  every  position  casts  a  circular  shadow 
must  be  a  sphere. 

4.  Diurnal  motion  of  the  heavenly  bodies. — Two  things  will  be 
noticed  by  an  observer  who  watches  the  heavens  during  any 
clear  night.  The  first  is,  that  all  the  heavenly  bodies,  with  the 
exception  of  the  moon  and  the  planets,  retain  constantly  the 
same  relative  situation  ;  and  the  second  is,  that  all  these  bodies, 
without  any  exception  whatever,  are  continually  changing  their 
positions  with  reference  to  the  horizon.  Let  us  suppose  the 
observer  to  be  at  some  place  in  the  Northern  Hemisphere.  A 
plane  passed  tangent  to  the  earth's  surface  at  his  feet  will  be  his 


Fig.  2. 


sensible   horizon,  and   a  second  plane,  parallel   to   this,  passed 
through  the  centre  of  the  earth,  will  be  his  rational  horizon.     If 


14  DIURNAL    MOTION. 

these  two  planes  oe  indefinitely  extended  in  every  direction,  they 
will  intersect  the  surface  of  the  celestial  sphere  in  two  circles; 
but  the  radius  of  the  celestial  sphere  is  so  immense  in  compari- 
son with  the  radius  of  the  earth,  that  these  two  circles  will 
sensibly  coincide,  and  will  form  one  great  circle  of  the  sphere, 
called  the  celestial  horizon.  In  other  words,  the  earth,  when 
compared  with  the  celestial  sphere,  is  to  be  regarded  as  only  a 
point  at  its  centre. 

Let  Fig.  2,  then,  represent  the  celestial  sphere,  at  the  centre 
of  which,  0,  the  observer  is  stationed.  Let  the  circle  HESW  be 
his  celestial  horizon,  of  which  His  the  north  point,  S  the  south, 
E  the  east,  and  W  the  west. 

If  he  looks  towards  the  southern  point  of  the  horizon,  and 
watches  the  movements  of  some  star  which  rises  a  little  east 
of  south,  he  will  see  that  it  rises  above  the  horizon  in  a  cir- 
cular path  for  a  little  distance,  attains  its  greatest  elevation 
above  the  horizon  when  it  bears  directly  south,  and  then 
descends  and  passes  below  the  horizon  a  little  west  of  south. 
In  the  figure,  abc  represents  the  path  of  such  a  star.  If  he 
notices  a  star  which  rises  more  to  the  eastward,  as  at  d,  for 
instance,  he  will  see  that  it  also  passes  from  the  east  quarter  of 
the  horizon  to  the  west  in  a  circular  path,  attaining,  however,  a 
greater  altitude  above  the  horizon  than  that  which  the  star  a 
attained,  and  remaining  above  the  horizon  a  longer  time.  A 
star  which  rises  in  the  east  point  will  set  in  the  west  point,  and 
will  remain  twelve  hours  above  the  horizon.  Turning  his  atten- 
tion to  some  star  which  rises  between  the  north  and  the  east,  as 
the  star  g,  for  instance,  he  will  find  that  its  movements  are  simi- 
lar to  the  movements  of  the  stars  already  noticed,  and  that  it 
will  remain  above  the  horizon  for  more  than  twelve  hours. 
Finally,  if  he  turns  towards  the  north,  he  will  see  certain  stars, 
called  circumpolar  stars,  which  never  pass  below  the  horizon,  but 
continually  revolve  about  a  fixed  point  in  the  heavens,  very  near 
to  which  point  is  a  bright  star  called  the  Pole-star,  which,  to  the 
naked  eye,  appears  to  be  stationary,  though  observation  shows 
that  it  also  revolves  about  this  same  fixed  point.  In  the  figure, 
Im  represents  the  orbit  of  a  circumpolar  star. 

If  the  same  course  of  observations  be  repeated  on  the  follow- 


DIURNAL    MOTION.  15 

ing  night,  the  observer  will  find  that  the  situations  of  the  stars 
with  reference  both  to  each  other  and  to  the  horizon  are  the 
same  that  they  were  when  he  first  began  to  examine  them  ;  the 
motions  which  have  already  been  described  will  be  repeated,  the 
circumpolar  stars  will  still  revolve  about  the  same  point  in  the 
heavens,  and,  in  short,  all  the  phenomena  of  which  he  took 
note  will  again  be  exhibited. 

5.  Inferences. — Three  important  truths  are  proved  by  a  series 
of  observations  similar  to  that  which  we  suppose  to  have  been 
made. 

(1.)  The  points  at  which  the  path  of  each  star  intersects  the 
horizon  remain  unchanged  from  night  to  night,  as  long  as  the 
geographical  position  of  the  observer  remains  the  same. 

(2.)  All  the  stars,  whether  they  move  in  great  or  in  small 
circles,  make  a  complete  revolution  in  identically  the  same  in- 
terval of  time; — that  is  to  say,  in  twenty-four  hours. 

(3.)  If  we  call  that  point  about  which  the  circumpolar  stars 
appear  to  revolve  the  north  pole  of  the  heavens,  and  call  the 
right  line  drawn  from  this  point  through  the  common  centre  of 
the  earth  and  the  celestial  sphere  the  axis  of  the  celestial  sphere, 
the  planes  of  the  circles  of  all  the  stars  are  perpendicular  to  this 
axis. 

Whether,  then,  the  earth  remains  at  rest,  and  the  celestial 
sphere  rotates  about  its  axis,  as  above  defined,  or  the  celestial 
sphere  remains  at  rest,  and  the  earth  rotates  within  it  on  an  axis 
of  its  own,  one  thing  is  certain :  the  axis  of  rotation  preserves 
in  either  case  a  constant  direction.  If  the  celestial  sphere 
rotates  about  its  axis,  this  axis  always  passes  through  the  same 
points  of  the  earth's  surface ;  and  if  the  earth  rotates  within  the 
celestial  sphere,  its  axis  of  rotation  is  constantly  directed  to  the 
same  points  on  the  surface  of  the  celestial  sphere. 

We  shall  see  hereafter  the  reasons  which  have  led  to  the 
adoption  of  the  theory  that  these  apparent  motions  of  the  stars 
are  really  due  to  the  rotation  of  the  earth  upon  its  own  axis. 
At  present,  for  the  sake  of  convenience  in  description,  we  shall 
Consider  the  earth  to  be  at  rest,  and  shall  speak  of  the  apparent 
motions  of  the  heavenly  bodies  as  though  they  were  real. 

6.  Farther  observations. — Let  us  now  suppose  that  the  observer 


16  DIURNAL   MOTION. 

leaves  the  place  where  he  has  hitherto  been  stationed,  and  travels 
in  the  direction  of  the  point  about  which  the  circumpolar  stars 
have  appeared  to  revolve,  and  which  we  have  called  the  north 
pole  of  the  heavens.  The  general  character  of  the  phenomena 
which  he  observes  will  not  be  changed ;  but  he  will  notice  a 
change  in  this  respect :  the  elevation  of  the  north  pole  above 
the  horizon  will  continually  increase  as  he  travels  towards  it, 
and  the  planes  of  the  circles  of  the  stars,  remaining  constantly 
perpendicular  to  the  axis  of  the  celestial  sphere,  will  become 
less  and  less  inclined  to  the  plane  of  the  horizon.  The  conse- 
quence of  this  will  be  that  stars  which  are  near  the  southern 
point  of  the  horizon  will  remain  a  shorter  time  above  the  hori- 
zon, and  will  finally  cease  to  appear ;  while  in  the  northern 
quarter  of  the  heavens  the  number  of  stars  which  never  pass 
below  the  horizon  will  continually  increase.  Finally,  if  we  sup- 
pose the  observer  to  go  on  until  the  north  pole  is  directly  above 
his  head,  the  stars  which  he  sees  will  neither  set  nor  rise,  but 
will  continually  move  about  the  sphere  in  circles  whose  planes 
are  parallel  to  the  plane  of  the  horizon.  In  such  a  situation,  it 
is  evident  that  half  of  the  celestial  sphere  will  be  perpetually 
invisible  to  him.  Referring  to  Fig.  2,  such  a  state  of  things  is 
represented  by  supposing  the  line  OP  to  be  moved  up  into  coin- 
cidence with  OZ,  the  planes  of  the  circles  abe,  def,  &c.,  still  re- 
maining perpendicular  to  OP.  The  stars  a  and  d  will  lie  con- 
tinually below  the  horizon,  and  the  stars  g  and  I  continually 
above  it. 

Such  a  sphere  as  this  just  now  described,  where  the  planes  of 
revolution  are  parallel  to  the  plane  of  the  horizon,  is  called  a 
parallel  sphere. 

If  the  observer,  instead  of  travelling  towards  the  north  pole, 
travels  directly  from  it,  its  elevation  above  the  horizon  will  con- 
tinually decrease,  and  the  obliquity  of  the  planes  of  the  circles 
to  the  plane  of  the  horizon  will  continually  increase,  until  he 
will  at  length  reach  a  point  at  which  the  north  pole  will  lie  in 
his  horizon  and  the  planes  of  ths  circles  will  be  perpendicular  to 
its  plane.  Referring  again  to  Fig.  2,  the  line  PO  will  in  this 
case  coincide  with  OH,  and  the  arcs  abc,  def,  &c.,  will  have 
their  planes  perpendicular  to  the  plane  of  the  horizon.  It  is 


.DEFINITIONS.  17 

evident  that  at  this  point  every  star  in  the  celestial  sphere  will 
come  above  the  horizon  once  in  twenty-four  hours,  and  that  half 
of  every  circle  will  lie  above  the  horizon,  and  half  below  it. 

The  geographical  position  which  the  observer  has  now  reached 
is  some  point  on  the  earth's  equator  (Art.  7).  Such  a  sphere  as 
this,  where  the  planes  of  the  circles  are  perpendicular  to  the  plane 
of  the  horizon,  is  called  a  right  sphere.  Besides  the  right  and 
the  parallel  sphere,  we  have  also  the  oblique  sphere,  where  the 
planes  of  the  circles  are  oblique  to  the  plane  of  the  horizon. 
Such  a  sphere  is  represented  in  Fig.  2. 

If  the  observer  travels  still  farther  in  the  same  direction,  the 
north  pole  will  sink  below  his  horizon,  and  the  other  extremity 
of  the  axis  of  the  celestial  sphere,  called  the  south  pole,  will 
rise  above  it.  There  will  be  circumpolar  stars  revolving  about 
this  pole,  and,  in  brief,  all  the  phenomena  which  the  observer 
noticed  while  travelling  towards  the  north  pole  will  be  repeated 
as  he  travels  towards  the  south  pole. 


DEFINITIONS. 

7.  We  are  now  prepared  to  define  certain  points,  angles,  and 
circles  on  the  earth  and  on  the  celestial  sphere. 

The  axis  of  the  celestial  sphere  is,  as  we  have  already  seen,  an 
imaginary  line  drawn  from  the  north  pole  of  the  heavens  through 
the  common  centre  of  the  earth  and  of  the  celestial  sphere, 
and  produced  until  it  again  meets  the  surface  of  the  celestial 
sphere  in  the  south  pole.  The  points  where  this  axis  meets  the 
surface  of  the  earth  are  called  the  north  pole  and  the  south  pole 
of  the  earth. 

That  pole  of  the  heavens  which  is  above  the  horizon  at  any 
place  is  called  the  elevated  pole  at  that  place;  the  other  is  called 
the  depressed  pole. 

The  axis  of  the  earth  is  that  diameter  which  passes  through 
the  poles  of  the  earth. 

Thf  earth's  equator  is  a  great  circle  of  the  earth,  whose  plane 
is  perpendicular  to  the  axis.  This  circle  is,  of  course,  equidis- 
tant from  the  two  poles,  and  divides  the  earth  into  two  hemi- 
spheres.  That  hemisphere  which  contains  the  north  pole  is 


18  DEFINITIONS. 

called  the  northern  hemisphere,  the  other  is  called  the  southern 
hemisphere. 

Parallels  of  latitude  are  small  circles  of  the  earth  whose  planes 
tire  perpendicular  to  the  axis. 

Terrestrial  meridians  are  great  circles  of  the  earth  passing 
tli rough  the  poles. 

The  latitude  of  any  place  on  the  earth's  surface  is  its  angular 
distance  from  the  plane  of  the  equator.  This  angle  is  measured 
by  the  arc  of  the  meridian  included  between  the  place  and  the 
equator.  Latitude  is  reckoned,  either  north  or  south,  from  0° 
to  90°. 

The  longitude  of  any  place  is  the  inclination  of  its  own  meri- 
dian to  the  meridian  of  some  fixed  station,  and  is  measured  by 
the  arc  of  the  equator  included  between  these  two  meridians. 
Longitude  is  usually  reckoned  east  or  west  of  the  fixed  meri- 
dian, from  (P  to  180°.  The  meridian  of  Greenwich,  England, 
is  most  commonly  taken  for  the  fixed  meridian,  though  the 
meridians  of  Washington,  Paris,  and  other  places  are  also  taken 
for  the  same  purpose.  The  fixed  meridian  is  called  the  prime 
meridian. 

The  arc  of  a  parallel  of  latitude  included  between  any  two 
meridians  is  the  departure  between  those  meridians  for  that  lati- 
tude. It  is  evident  that  if  the  departure  between  any  two  meri- 
dians is  taken  on  two  different  parallels  of  latitude,  the  depart- 
ure at  the  greater  latitude  will  be  the  smaller. 

Upward  motion  at  any  place  is  motion  from  the  centre  of  tho 
earth.  Downward  motion  is  towards  the  same  point. 

The  celestial  horizon  has  already  been  defined  (Art.  4).  Lines 
drawn  perpendicular  to  the  plane  of  the  horizon  are  called 
vertical  lines.  The  vertical  line  at  any  place,  if  indefinitely 
prolonged,  meets  the  celestial  sphere  in  two  points.  The  upper 
point  is  called  the  zenith,  the  lower  point  the  nadir. 

The  celestial  meridian  of  any  place  is  the  great  circle  in  which 
the  plane  of  the  terrestrial  meridian  of  that  place,  when  indefi- 
nitely produced,  meets  the  surface  of  the  celestial  sphere.  The 
axis  of  the  sphere  divides  the  celestial  meridian  into  two  semi- 
circumferences  :  that  which  lies  on  the  same  side  of  the  axis  as 
the  zenith  is  called  the  upper  branch  of  the  meridian,  und  tho 


DEFINITIONS.  19 

other  is  called  the  lower  branch.  The  points  where  the  celestial 
meridian  and  the  celestial  horizon  intersect  are  called  the  north 
and  the  south  point  of  the  horizon, — the  point  which  is  the  nearer 
to  the  north  pole  being  the  north  point.  The  line  in  which 
the  planes  of  these  same  two  great  circles  intersect  is  called  the 
meridian  line. 

Vertical  circles  are  great  circles  of  the  sphere  which  pass 
through  the  zenith  and  nadir.  That  vertical  circle  the  plane 
of  which  is  perpendicular  to  the  plane  of  the  celestial  meridian 
is  called  the  prime  vertical.  The  points  in  which  the  prime  ver- 
tical cuts  the  horizon  are  called  the  east  and  the  west  point  of  the 
horizon ;  and  the  line  which  joins  these  two  points  is  the  east 
and  west  line. 

The  celestial  equator,  also  called  the  equinoctial,  is  the  great 
circle  of  the  sphere  in  which  the  plane  of  the  earth's  equator, 
indefinitely  produced,  meets  the  celestial  sphere. 

The  altitude  of  a  heavenly  body  is  its  angular  distance  above 
the  plane  of  the  celestial  horizon,  measured  on  a  vertical  circle 
passing  through  that  body.  The  zenith  distance  of  the  body  is 
its  angular  distance  from  the  zenith,  and  is  evidently  the  com- 
plement of  the  altitude. 

The  azimuth  of  a  celestial  body  is  the  inclination  of  the  verti- 
cal circle  which  passes  through  the  body  to  the  celestial  meri- 
dian, and  is  measured  by  the  arc  of  the  celestial  horizon  in- 
cluded between  this  vertical  circle  and  the  celestial  meridian. 
Azimuth  may  be  reckoned  from  either  the  north  or  the  south 
point  of  the  horizon,  and  towards  either  the  west  or  the  east. 
Navigators  usually  reckon  it  from  the  north  point  in  north  lati- 
tude and  the  south  point  in  south  latitude,  and  east  or  west  as 
the  body  is  east  or  west  of  the  meridian,  thus  restricting  it 
numerically  to  values  less  than  180°.  Astronomers,  on  the  con- 
trary, usually  reckon  from  the  south  point,  to  the  right  hand, 
from  0°  to  36(T. 

The  amplitude  of  a  heavenly  body  is  its  angular  distance  from 
the  prime  vertical  when  in  the  horizon.  It  is  reckoned  from 
the  east  point  when  the  body  is  rising,  and  from  the  west  point 
when  the  body  is  setting,  towards  the  north  or  the  south  as  the 
hotly  i&  to  the  north  or  the  south  or  the  prime  vertical. 


20  DEFINITIONS. 

The  vernal  equinox  is  a  certain  fixed  point  upon  the  equinoc- 
tial. It  is  also  called  the  first  point  of  Aries. 

Hour  circles,  or  circles  of  declination,  are  great  circles  of  the 
sphere  passing  through  the  poles  of  the  heavens. 

The  right  ascension  of  a  heavenly  body  is  the  inclination  of 
its  hour  circle  to  the  hour  circle  which  passes  through  the  vernal 
equinox ;  or  it  is  the  arc  of  the  equinoctial  intercepted  between 
these  two  hour  circles.  Right  ascension  is  usually  reckoned  in 
hours,  minutes,  and  seconds  (an  hour  being  taken  equal  to  15° 
of  arc),  and  is  always  reckoned  to  the  eastward,  from  Oh.  to  24h. 

The  declination  of  a  heavenly  body  is  its  angular  distance 
from  the  plane  of  the  celestial  equator,  measured  on  the  circle 
of  declination  passing  through  the  body.  It  is  reckoned  in 
degrees,  minutes,  &c.,  to  the  north  and  the  south.  The  polar 
distance  of  a  body  is  its  angular  distance  from  either  pole,  mea- 
sured on  its  hour  circle.  Usually,  however,  when  we  speak  of 
the  polar  distance  of  a  body,  we  mean  its  angular  distance  from 
the  elevated  pole. 

If,  in  Fig.  2,  with  the  arc  HP,  which  measures  the  altitude 
of  the  elevated  pole,  as  a  polar  radius,  we  describe  a  circle  about 
the  pole  as  a  centre,  it  is  evident  that  the  stars  whose  circles  lie 
within  this  circle  will  never  set.  This  circle  is  called  the  circle 
of  perpetual  apparition.  It  is  equally  evident  that  stars  whose 
circles  lie  within  a  circle  of  the  same  magnitude,  described  about 
the  depressed  pole  as  a  centre,  will  never  come  above  the  horizon. 
This  circle  is  therefore  called  the  circle  of  perpetual  occupation. 

The  passage  of  a  celestial  body  across  the  meridian  is  called 
its  transit  or  culmination.  When  the  body  is  within  the  circle 
of  perpetual  apparition,  both  transits  occur  above  the  horizon, 
one  above  the  pole,  the  other  below  it.  These  are  called  the 
upper  and  the  lower  transit.  For  all  bodies  outside  this  circle, 
and  not  within  the  circle  of  perpetual  occultation,  the  upper 
transit  occurs  above  the  horizon,  the  lower  below  it.  For  all 
bodks  whatever,  the  upper  transit  occurs  when  the  body  crosses 
the  upper  branch  of  the  meridian,  and  the  lower  transit  when 
it  crosses  the  lower  branch. 

The  hour  angle  of  a  heavenly  body  is  the  inclination  of  the 
circle  of  declination  which  passes  through  the  body  to  the 


CELESTIAL   SPHERE. 


21 


celestial  meridian,  and  is  measured  by  the  arc  of  the  celestial 
equator  included  between  these  two  circles.  Hour  angles  are 
reckoned  positively  towards  the  west,  from  the  upper  culmina- 
tion, from  0°  to  360°,  or  Oh.  to  24h. 

The  hour  angle  of  the  sun  is  called  solar  time,  and  that  of  the 
first  point  of  Aries  sidereal  time.  The  interval  of  time  between 
two  consecutive  upper  transits  of  the  sun  is  called  a  solar  day, 
and  the  interval  between  the  upper  transits  of  the  first  point  of 
Aries  is  called  a  sidereal  day.  The  celestial  sphere  apparently 
makes  one  revolution  about  the  earth  in  a  sidereal  day.  The 
solar  day  is,  on  the  average,  about  3m.  56s.  longer  than  the 
sidereal  day. 

8.  Some  of  the  preceding  definitions  are  illustrated  in  the 
diagram,  Fig.  3.  In  this  figure  0  is  the  position  of  the  observer, 
HESW  his  celestial  horizon,  Pp  the  axis  of  the  heavens,  P 
the  elevated  and  p  the  depressed  pole,  Zthe  zenith,  and  A^the 
nadir.  The  circle  HZSN  is  the  observer's  celestial  meridian. 
It  may  be  noticed  that  this  circle  is  at  once  a  vertical  and  an 


hour  circle.     The  circle  ECWD  is  the  equinoctial,  and  the  circle 
EZWN,  perpendicular  to  the  meridian,  is  the  prime  vertical, 


22  THIXJKEMS. 

cutting  the  horizon  in  E  and  W^  the  east  and  the  west  point  of 
the  horizon.  The  equinoctial,  being  also  perpendicular  to  the 
meridian,  passes  through  the  same  points.  If  P  is  supposed  to 
be  the  north  pole,  if  is  the  north  point  of  the  horizon,  and  8  the 
south  point. 

Let  A  denote  some  celestial  body.  GA  is  its  altitude,  ZA  its 
zenith  distance,  H G  its  azimuth  as  reckoned  by  navigators,  and 
SG  its  azimuth  as  reckoned  by  astronomers,  all  these  elements 
of  position  being  determined  by  the  arc  of  a  vertical  circle,  ZG, 
passed  through  A.  Let  an  arc  of  an  hour  circle,  PB,  be  also 
passed  through  A.  Then  is  AB  its  declination,  PA  its  polar 
distance,  and  if  V  be  taken  to  denote  the  position  of  the  vernal 
equinox,  VB  is  the  right  ascension  of  A.  The  right  ascension 
may  also  be  represented  by  the  angle  VPS,  which  the  arc  VB 
measures.  The  angle  ZPB  is  the  hour  angle  of  A,  and  ZPV  is 
the  hour  angle  of  the  vernal  equinox,  or  the  sidereal  time. 
This  angle  may  also  be  designated  as  the  right  ascension  of  the 
meridian. 

The  circle  KH,  drawn  about  P  with  the.  radius  PH,  is  the 
circle  of  perpetual  apparition.  The  star  whose  path  is  repre- 
sented by  Im  never  passes  below  the  horizon :  I  is  its  upper,  m 
its  lower  culmination.  SR  represents  the  circle  of  perpetual 
occultation,  and  the  stars  whose  paths  lie,  like  or,  within  this 
circle,  never  come  above  the  horizon. 

9.  Theorem. — The  sidereal  time  at  any  place  is  always  equal  to 
the  sum  of  the  right  ascension  and  the  hour  angle  of  the  same  body. 
This  is  an. important  astronomical  theorem,  and  is  readily  proved 
in  Fig.  3,  in  which  ZPV,  the  sidereal  time,  is  the  sum  of  ZPB, 
the  hour  angle,  and  VPB,  the  right  ascension  of  the  celestial 
body  A.  Any  two  of  these  three  angles,  then,  being  given,  the 
third  is  readily  obtained. 

Corollary. —  When  a  celestial  body  is  at  its  upper  culmination 
at  any  place,  its  right  ascension  is  equal  to  the  sidereal  time  at  that 
place.  This  is  evidently  true,  because  the  hour  angle  of  a  body 
when  at  its  upper  culmination  is  zero,  and  hence,  from  the  theorem, 
the  right  ascension  of  the  body  is  equal  to  the  sidereal  time.  For 
example,  in  Fig.  3,  if  I  is  a  body  at  its  upper  culmination,  VPZ  is 
both  its  right  ascension  and  the  right  ascension  of  the  meridian. 


THEOREMS. 


Fig.  4. 


10.  Theorem. —  The  latitude  of  any  place  on  the  earth's  surface 
is  equal  to  the  altitude  of  the  elevated  pole  at  that  place.     Let  L 
(Fig.  4)  be  some   place   on  the 

earth's  surface,  Pp  the  earth's 
axis,  and  EQ  the  equator.  The 
line  HRy  tangent  to  the  earth's 
surface  at  L,  is  the  horizon,  and 
Z  the  zenith,  of  L.  According  to 
the  definition  already  given,  LOQ 
is  the  latitude  of  L.  Let  the 
earth's  axis  be  indefinitely  pro- 
longed, and  at  L  let  the  line  LP", 
parallel  to  the  earth's  axis,  be  also 
indefinitely  prolonged.  Owing 
to  the  immensity  of  the  celestial  sphere  when  compared  with 
the  earth,  these  two  lines  will  sensibly  meet  at  a  common  point 
on  the  surface  of  the  celestial  sphere,  and  this  common  point 
will  be  the  elevated  pole.  The  elevated  pole,  then,  to  an  ob- 
server at  L,  will  lie  in  the  direction  LP",  and  P'LH  will  be  its 
altitude. 

Now  we  have,  HLZ  =  POQ 

ZLP"  =  ZOPf 
.'.P"LH=LOQ 
which  was  to  be  proved. 

11.  Theorem. —  The  latitude  of  a  place  is  equal  to  the  declination 
of  the  zenith  of  that  place.     This  is  easily  seen  in  Fig.  4 :  for  the 
declination  of  any  body  or  point  is  its  angular  distance  from  the 
plane  of  the  celestial  equator,  and  hence  ZOQ  in  the  declination 
of  the  zenith. 

Either  of  these  theorems  might  be  deduced  from  the  other, 
but  it  is  better  to  consider  them  as  independent  propositions. 

12.  The  Astronomical  Triangle. — The  spherical  triangle  PZA 
(Fig.  3)  is  called  the  astronomical  triangle.     It  is  formed  by  the 
arcs  of  the  meridian  of  the  -place,  and  of  the  vertical  circle  and 
the  hour  circle  passing  through  some  heavenly  body,  which  are 
included  between  the  zenith  of  the  observer,  the  elevated  pole, 
and  the  position  of  the  body  as  projected  on  the  surface  of  the 
celestial  sphere.     The  three  sides  are:   %P,  the  co-latitude  of  the 


24  DIURNAL   CIRCLES. 

place;  PA,  the  polar  distance  of  the  body;  and  ZA,  its  zenith 
distance.  The  three  angles  are:  ZPA,  the  hour  angle  of  the 
body;  PZA,  its  azimuth;  and  PAZ,  an  angle  which  is  rarely 
used,  and  which  is  commonly  called  the  position  angle  of  the 
body. 

The  co-latitude  and  the  zenith  distance  can  evidently  never 
be  greater  than  90°.  The  polar  distance  is  equal  to  90°  minm 
the  declination:  and  it  is  less  than  90°  when  the  body  is  on  the 
same  side  of  the  celestial  equator  as  the  observer,  but  becomes 
numerically  90°  plus  the  declination  when  the  body  is  on  the 
opposite  side.  In  the  former  case  the  declination  of  the  body 
is  said  to  have  the  same  name  as  the  latitude,  in  the  latter  case, 
to  have  the  opposite  name. 

13.  Diurnal  Circles. — We  have  already  seen  that  the  apparent 
daily  motions  of  the  stars  are  performed  in  circles,  the  planes 
of  which  are  perpendicular  to  the  axis  of  the  sphere.  These 
circles  are  called  diurnal  circles.  The  phenomena  which  have 
been  observed  with  reference  to  these  circles  (Arts.  4  and  6)  are 
explained  in  Fig.  3.  The  circles  of  all  the  stars  which  rise  in 
the  arc  of  the  horizon,  ES,  are  evidently  divided  by  the  horizon 
into  two  unequal  parts,  the  smaller  of  which  in  each  case  lies  above 
the  horizon.  Hence  these  stars  are  above  the  horizon  less  than 
twelve  hours.  On  the  other  hand,  the  greater  portions  of  the 
circles  of  those  stars  which  rise  between  E  and  H  lie  above  the 
horizon,  and  the  stars  themselves  are  above  it  for  more  than 
twelve  hours. 

Any  body,  then,  whose  declination  is  of  the  same  name  as  the 
elevated  pole  will  be  above  the  horizon  more  than  twelve  hours, 
while  a  body  whose  declination  is  of  a  different  name  from  that 
of  the  elevated  pole  will  be  above  the  horizon  less  than  twelve 
hours.  And  further:  those  bodies  which  are  in  north  declination, 
in  other  words,  to  the  north  of  the  celestial  equator,  will  rise  to 
the  north  of  east  and  set  to  the  north  of  west;  while  bodies  in 
south  declination  will  rise  and  set  to  the  south  of  the  east  and 
the  west  point. 

When  a  star  has  no  declination,  that  is  to  say,  is  on  the  celestial 
equator,  it  will  rise  due  east  and  set  due  west,  and  will  remain 
twelve  hours  above  the  horizon. 


SPHERICAL    CO-ORDINATES.  25 

1-1  Right  and  Parallel  /Spheres. — When  an  observer  travels 
towards  the  elevated  pole,  the  radius  of  the  circle  of  perpetual 
apparition,  being  equal  to  the  altitude  of  the  pole,  continually 
increases,  and  the  number  of  stars  which  never  set  increases  in 
like  manner.  The  number  of  stars  which  never  rise  also  increases. 
Finally,  if  he  reaches  the  pole,  the  celestial  equator  coincides 
with  the  horizon,  the  east  and  the  west  point  disappear,  and  the 
bodies  which  are  on  the  same  side  of  the  equator  with  the  ob- 
server are  perpetually  above  the  horizon,  and  revolve  in  circles 
whose  planes  are  parallel  to  it,  while  the  bodies  which  are  on 
the  opposite  side  of  the  equator  never  rise.  As  he  travels  to- 
wards the  equator,  the  circles  of  perpetual  apparition  and  occul- 
tation  alike  diminish,  the  diurnal  circles  become  more  and  more 
nearly  vertical,  and  when  he  reaches  the  equator,  the  equinoctial 
becomes  perpendicular  to  the  horizon  and  coincides  with  the 
prime  vertical,  and  the  horizon  bisects  all  the  diurnal  circles. 
At  the  equator,  then,  every  celestial  body  comes  above  the  hori- 
zon, and  remains  above  it  twelve  hours. 

15.  Spherical  Co-ordinates. — The  position  of  any  point  on  the 
surface  of  a  sphere  is  determined,  as  soon  as  its  angular  distances 
are  given   from  any  two  great  circles  on  that  sphere  whose  po- 
sitions are  known.     Thus  the  geographical  position  of  any  point 
on  the  earth's  surface  is  known  when  we  have  determined  its 
latitude  and  longitude;  in  other  words,  when  we  know  its  an- 
gular distance  from  the  equator  and  from  the  prime  meridian. 
In  like  manner  we  know  the  position  of  any  point  on  the  sur- 
face of  the  celestial  sphere  when  either  its  altitude  and  azimuth, 
or  its  right  ascension  and  declination,  are  given.     In  the  first 
of  these  two  systems  of  co-ordinates  the  fixed  great  circles  are 
the  celestial  horizon  and  the  celestial  meridian,  the  origin  of  co- 
ordinates being  either  the  north  or  the  south  point  of  the  horizon. 
In  the  second  system  the  fixed  great  circles  are  the  equinoctial 
and  the  circle  of  declination  which  passes  through  the  vernal 
equinox,  and  the  origin  of  co-ordinates  is  the  vernal  equinox. 
In  Fig.  3,  if  we  know  the  arcs  HG  and   GA,  or  the  arcs  VB 
and  BA,  we  evidently  know  the  position  of  A. 

16.  Vanishing  Points  and  Vanishing  Circles. — Every  one  knows- 
that  as  h?  increases  the  distance  between  himself  and  any  object, 


26  VANISHING    POINTS    AND    CIRCLES. 

the  apparent  magnitude  of  the  object  decreases ;  and  that,  if  he 
recedes  far  enough  from  it,  it  witl  be  reduced  in  appearance  to 
a  point.  Every  one  also  knows  that  when  he  looks  along  the  line 
of  a  railroad  track  the  lines  appear  to  converge,  and  that,  if 
the  track  is  straight,  and  the  curvature  of  the  earth  does  not 
limit  his  vision,  the  rails  will  ultimately  appear  to  meet. 

These  familiar  illustrations  will  serve  to  show  what  is  meant 
by  a  vanishing  point.  The  actual  distance  between  the  rails  of 
course  remains  the  same ;  but  the  angle  at  the  eye  which  this 
distance  subtends  decreases  as  the  eye  is  directed  along  the 
track,  until  at  last  it  ceases  to  subtend  any  appreciable  angle  at 
the  eye,  and  the  rails  apparently  meet.  This  point  where  the 
rails  appear  to  meet  is  called  the  vanishing  point  of  the  two 
lines;  and,  in  general,  the  vanishing  point  of  any  system  of  pa- 
rallel lines  is  the  point  at  which  they  will  appear  to  meet,  when 
indefinitely  prolonged.  We  have  already  seen  (Art.  10)  that 
the  pole  of  the  heavens  is  the  vanishing  point  of  lines  drawn 
perpendicular  to  the  equator,  and  the  same  may  be  said  of  the 
poles  of  any  circle  on  the  celestial  sphere.  For  instance,  the 
poles  of  the  horizon  at  any  place  are  the  zenith  and  the  nadir; 
and  any  system  of  lines  perpendicular  to  the  horizon  will  appa- 
rently meet,  when  prolonged  indefinitely,  in  these  two  points. 
And  again,  the  east  and  the  west  point  of  the  horizon  are  the 
poles  of  the  meridian,  and  lines  drawn  perpendicular  to  the 
meridian  will  have  these  points  for  their  vanishing  points. 

The  same  principle  holds  good  when  applied  to  any  system 
of  parallel  planes.  They  will  appear  to  meet,  when  indefinitely 
extended,  in  one  great  circle  of  the  sphere,  and  this  circle  is 
called  the  vanishing  circle  of  that  system  of  planes.  The  celes- 
tial horizon  is,  as  has  already  been  stated  (Art.  4),  the  vanishing 
circle  of  the  planes  of  the  sensible  and  the  rational  horizon,  and 
indeed  of  any  number  of  planes  passed  parallel  to  them.  The 
celestial  equator  is  the  vanishing  circle  of  the  planes  of  all  the 
parallels  of  latitude,  and,  in  short,  every  circle  of  the  celestial 
sphere  may  be  regarded  as  the  vanishing  circle  of  a  system  of 
planes  passed  perpendicular  to  the  line  which  joins  the  poles  of 
that  circle. 

17.  Spherical  Prnject'&ns. — The  point?  and  circles  of  either  the 


SPHERICAL    PROJECTIONS.  27 

earth  or  the  celestial  sphere,  or  of  both,  may  be  projected  upon 
the  plane  of  any  great  circle  of  either  sphere.  The  plane  on 
which  the  projections  are  made  is  called  the  primitive  plane,  and 
the  circle  which  bounds  this  plane  is  called  the  primitive  circle. 
Several  distinct  methods  of  projection  will  be  found  in  treatises 
on  Descriptive  Geometry.  Of  these,  the  most  common  are  the 
orthographic,  the  stereographic,  and  Mercator's  projection. 

In  the  orthographic  projection,  the  point  of  sight  is  taken  in 
the  axis  of  the  primitive  circle,  and  at  an  infinite  distance  from 
that  circle.  All  circles  whose  planes  are  perpendicular  to  the 
primitive  plane  are  projected  into  right  lines;  all  circles  whose 
planes  are  parallel,  to  the  primitive  plane  are  projected  into 
circles,  each  of  which  is  equal  to  the  circle  of  which  it  is  a  pro- 
jection ;  and  all  other  circles  are  projected  into  ellipses. 

In  the  stereographic  projection,  the  point  of  sight  is  at  either 
pole  of  the  primitive  circle,  and  its  distance  from  that  circle  is 
finite.  In  this  projection  every  circle  is  projected  as  a  circle, 
unless  its  plane  passes  through  the  point  of  sight,  in  which  case 
it  is  projected  into  a  right  line. 

Mercator's  projection  is  employed  in  the  construction  of  charts 
representing  the  earth's  projection.  In  this  projection  the 
parallels  of  latitude  are  represented  by  parallel  right  lines,  and 
the  meridians  are  also  represented  by  parallel  right  lines,  per- 
pendicular to  the  equator.  The  meridian  projections  are  equi- 
distant, but  the  distance  between  the  successive  latitude  projec- 
tions increases  as  we  recede  from  the  equator.  The  advantage 
offered  by  this  projection  to  navigators  is  that  the  ship's  track, 
as  long  as  the  course  on  which  it  sails  is  unaltered,  is  represented 
on  the  chart  by  a  straight  line,  and  that  the  angle  which  this 
line  makes  with  each  meridian  is  the  course, 


28  INSTRUMENTS. 


CHAPTER  II. 

ASTRONOMICAL   INSTRUMENTS.      ERRORS. 

18.  THIS  chapter  will  be  devoted  to  a  general  description  of 
the  common  astronomical  instruments,  of  the  class  of  observa- 
tions to  which  each  is  adapted,  and  of  the  manner  in  which  such 
observations  are  made.     No  attempt  will  be  made  to  describe 
the  elaborate  mechanism  by  which,  in  many  cases,  the  usefulness 
of  the  instrument  is  increased  and  its  manipulation  is  facilitated ; 
but  enough,  it  is  hoped,  will  be  said  to  enable  the  student  to 
form  a  clear  conception  of  the  prominent  features  of  each  instru- 
ment which  is  described.     There  is  no  lack  of  excellent  treatises 
on  Astronomy,  in  which  those  who  wish  to  investigate  this  subject 
more  thoroughly  will  find  all  the  details,  which  the  limits  pre- 
scribed to  this  book  will  not  permit  to  enter  here,  clearly  and 
elaborately  presented. 

THE    ASTRONOMICAL   CLOCK. 

19.  The  astronomical   clock   is  a  clock  which  is   regulated 
to  keep  sidereal  time,  and  is  an  indispensable  companion  to  the 
other  astronomical  instruments.     It  is  provided  with  a  pendulum 
so  constructed  that  change  of  temperature  will  not  affect  its 
length.     The  sidereal  day  at  any  place  commences,  as  has  al- 
ready been  stated,  when  the  vernal  equinox  is  on  the  upper 
branch  of  the  meridian  of  that  place,  and   the  theory  of  the 
sidereal  clock  is  that  it  shows  Oh.  Om.  Os.  when  the  vernal  equi- 
nox is  so  situated.     Practically,  however,  it  is  found  that  every 
clock  has  a  daily  rate;  that  is  to  say,  it  gains  or  loses  a  certain 
amount  of  time  daily.     In  order,  then,  that  a  clock  may  be  re- 
gulated to  sidereal  time,  it  is  necessary  to  know  both  its  error 
si nd  its  dally  rate;  the  error  being  the  amount  by  which  it  is 
fast  or  slow  at  any  given  time,  and    the  daily  rate  being  the 


THE    CLOCK.  29 

amount  which  it  gains  or  loses  daily;  and  knowing  these,  it 
is  evidently  in  our  power  to  obtain  at  any  desired  instant 
the  true  sidereal  time  from  the  time  shown  by  the  face  of  the 
clock. 

It  is  to  be  noticed  further,  that,  except  as  a  matter  of  con- 
venience, a  small  rate  has  no  advantage  over  a  large  one;  but 
it  is  very  important  that  the  rate,  whether  large  or  small,  shall 
be  constant  from  day  to  day;  so  that,  of  two  clocks,  one  of  which 
has  a  large  and  constant  rate,  and  the  other  a  small  and  varying 
one,  the  preference  is  to  be  given  to  the  former. 

Clocks  may  be  regulated  to  keep  either  local  sidereal  time  or 
Greenwich  sidereal  time,  or  both. 

20.  Error  of  the  Clock. — To  obtain  the  error  of  a  clock  on  the 
local  sidereal  time  at  any  observatory,  we  make  use  of  the  pro- 
position, already  demonstrated  (Art.  9),  that  the  right  ascension 
of  any  celestial  body,  when  at  its  upper  culmination  on  any 
meridian,  is  equal  to  the  sidereal  time  at  that  meridian.     The 
Nautical  Almanac  gives  the  right  ascensions  of  more  than  a 
hundred  stars  which  are  suitable  for  observations  for  time.     By 
means  of  an  instrument,  properly  adjusted,  we  determine  the 
instant  when  any  one  of  these  stars  is  on  the  meridian,  and  the 
time  which  the  clock  shows  at  that  instant  is  noted.     This  is  the 
clock  time  of  transit,  and  the  right  ascension  of  the  star,  taken 
from  the  Almanac,  is  the  true  time  of  transit;  and  a  comparison 
of  these  two  times  will  evidently  give  us  the  amount  by  which 
the  clock  is  fast  or  slow  on  local  sidereal  time. 

21.  Daily  Rate. — If,  in  a  similar  manner,  we  obtain  the  error 
of  the  clock  on  the  next  or  on  any  subsequent  day,  the  difference 
of  these  two  errors  will  be  the  gain  or  loss  of  the  clock  in  the 
interval;  and  hence,  if  we  divide  this  difference  by  the  number 
of  the  days  and  parts  of  days  which  have  intervened  between 
the  two  observations,  the  quotient  will   be  the  daily  gain  or 
loss. 

22.  Chronograph. — The  accuracy  of  astronomical  observations 
id  much  enhanced  by  recording  the  times  of  the  observations  by 
means  of  an  electric  current.     A  cylinder,  about  which  a  roll 
of  paper  is  wound,  is  turned  about  on  its  axis  with  a  uniform 
motion  by  the  use  of  appropriate  machinery.     A  pen  is  pressed 


30 


THE   TRANSIT   INSTRUMENT. 


Fig.  5. 


THE   TRANSIT   INSTRUMENT.  31 

down  upon  the  paper,  and  is  so  connected  with  a  battery  that 
whenever  the  circuit  is  broken  a  mark  of  some  kind  is  made  upon 
the  paper.  The  wires  of  this  battery  are  connected  with  the 
sidereal  clock  in  such  a  way  that  every  oscillation  of  the  pendu- 
lum breaks  the  circuit.  Every  second  is  thus  recorded  upon 
the  revolving  paper.  The  observer  also  holds  in  his  hands  a 
break-circuit  key,  with  which,  whenever  he  wishes  to  note  the 
time,  he  breaks  the  circuit,  and  thus  causes  the  pen  to  make  its 
mark  upon  the  paper.  The  line  which  the  pen  describes  upon 
the  paper  will  be  something  like  this : 

A  BO 

L i_l l_l _L_ 


The  equidistant  marks,  a,  b,  c,  &c.,  are  the  marks  caused  by  the 
pendulum,  and  the  marks  J,  B,  C,  are  the  marks  which  the  pen 
makes  when  the  circuit  is  broken  by  the  observer.  The  distance 
from  A  to  6,  B  to  c,  &c.,  can  be  measured  by  a  scale  of  equal 
parts,  and  the  time  of  an  observation  can  thus  be  obtained  within 
a  small  fraction  of  a  second. 

The  cylinder  is  also  moved  by  a  fine  screw  in  the  direction 
of  its  own  length,  so  that  the  pen  records  in  a  spiral. 

This  instrument  is  called  a  Chronograph.  There  are  several 
varieties  of  the  chronograph  in  use  by  different  astronomers,  but 
the  main  principle  in  all  of  them  is  similar  to  that  of  the  instru- 
ment just  now  described. 

THE   TRANSIT  INSTRUMENT. 

23.  The  transit  instrument  is  used,  as  its  name  implies,  in 
observing  the  transits  of  the  heavenly  bodies..  Fig.  5  represents 
a  transit  instrument.  It  consists  of  a  telescope,  TT,  sustained 
by  an  axis,  AA,  at  right  angles  to  it.  The  extremities  of  this 
axis  terminate  in  cylindrical  pivots,  which  rest  in  metallic  sup- 
ports, FF,  shaped  like  the  upper  part  of  the  letter  Y,  and  hence 
called  the  Ys.  These  Ys  are  imbedded  in  two  stone  pillars.  In 
order  to  relieve  the  pivots  of  the  friction  to  which  the  weight 
of  the  telescope  subjects  them,  and  to  facilitate  the  motion  of  the 


32  THE   TRANSIT   INSTRUMENT. 

telescope,  there  are  two  counterpoises,  WW,  connected  with 
levers,  and  acting  at  XX,  where*there  are  friction  rollers  upon 
which  the  axis  turns.  When  the  instrument  is  properly  adjusted, 
the  telescope,  as  it  turns  about  with  the  axis  AA,  will  continu- 
ally lie  in  the  plane  of  the  meridian ;  and,  in  order  to  effect  this, 
the  axis  A  A  should  point  to  the  east  and  the  west  point  of  the 
horizon,  and  be  parallel  to  its  plane.  There  are  therefore  screws 
at  the  ends  of  the  axis,  by  which  one  extremity  of  the  axis  may 
be  raised  or  depressed,  and  may  also  be  moved  forward  or 
backward. 

24.  The  Reticule. — In   the   common  focus 
of  the  object-glass  and  the  eye-glass  is  placed 
the  reticule,  a  representation  of  which  is  given 
in  Fig.  6.     It  consists  of  several  equidistant 
vertical  wires  (usually  seven)  and  two  hori- 
zontal ones.     If  the  instrument  is  accurately 

adjusted   to  the  plane  of  the  meridian,  the  ¥ig-  6- 

instant  that  any  star  is  on  the  middle  wire  is  the  instant  of  its 
transit.  These  wires  are  also  called  the  cross-wires. 

25.  Adjustment. — The  axis  of  rotation  of  the  instrument  is  an 
imaginary  line  connecting  the  central  points  of  the  pivots. 

The  axis  of  collimation  is  an  imaginary  line  drawn  from  the 
optical  centre  of  the  object-glass,  perpendicular  to  the  axis  of 
rotation. 

The  line  of  sight  is  an  imaginary  line  drawn  from  the  optical 
centre  of  the  object-glass  to  the  middle  wire. 

The  transit  instrument  is  accurately  adjusted  in  the  plane  of 
the  meridian,  when  the  line  of  sight  of  the  telescope  lies  con- 
tinually in  that  plane,  as  the  telescope  revolves.  Three  things, 
then,  are  readily  seen  to  be  necessary :  the  axis  of  rotation  must 
be  exactly  horizontal ;  it  must  lie  exactly  east  and  west ;  and  the 
line  of  sight  and  the  axis  of  collimation  must  exactly  coincide. 
Practically,  these  conditions  are  rarely  fulfilled;  but  they  can,  by 
repeated  experiments,  be  very  nearly  fulfilled,  and  the  errors 
which  the  failure  rigorously  to  adjust  the  instrument  causes  in 
•'.he  observations  will  be  constant  and  small,  and  can  be  accu- 
rately determined. 

20.  Application. — The   principal   application   of   the  transit 


THE   MERIDIAN    CIRCLE. 


33 


Fig  7 


34  THE    MERIDIAN    CIRCLE. 

instrument  in  observatories  is  to  the  determination  of  the  right 
ascensions  of  celestial  bodies.  Knowing  the  constant  instru- 
mental errors  just  now  mentioned,  and  the  error  and  the  rate  of 
the  clock,  we  can  easily  obtain  the  true  sidereal  time  at  the 
instant  of  the  transit  of  a  celestial  body,  which  time  is  at  once, 
as  we  have  already  seen,  the  right  ascension  of  that  body. 


THE    MERIDIAN    CIRCLE. 

27.  The  meridian  circle  is  a  combination  of  a  transit  instru- 
ment, similar  to  the  one  above  described,  and  a  graduated  circle, 
securely  fastened  at  right  angles  to  the  horizontal  axis,  and 
turning  with  it.  A  meridian  circle  which  is  set  up  at  the  United 
States  Naval  Academy,  Annapolis,  Maryland,  is  represented  in 
Fig.  7.  The  horizontal  axis  bears  two  graduated  circles,  CC, 
C'C'y  the  first  of  these  circles  being  much  more  finely  graduated 
than  the  second,  the  latter  being  only  used  as  a,  finder,  to  set  the 
telescope  approximately  at  any  desired  altitude.  R  and  R  re- 
present two  of  four  stationary  microscopes,  by  which  the  circle 
CCis  read;  LL  is  a  hanging  level,  by  which  the  horizontally  of 
the  axis  is  tested.  The  cross-wires  are  illuminated  by  light 
which  passes  from  a  lamp  through  the  tubes  AA,  and  through 
the  pivots  which  are  perforated  for  this  purpose,  and  is  reflected 
towards  the  reticule  by  a  metallic  speculum  which  is  set  within 
the  hollow  cube  M.  The  quantity  of  light  admitted  is  regulated 
by  revolving  discs  with  eccentric  apertures,  which  are  placed 
between  the  Ys  and  the  tubes  AA,  and  are  moved  by  cords  car- 
rying small  weights,  JSS. 

The  object  in  having  so  many  reading  microscopes  is  to  dim- 
inish the  errors  arising  from  the  imperfect  graduation  of  the  ver- 
tical circle.  When  any  angle  is  to  be  measured,  the  readings  of 
all  four  microscopes  are  first  taken.  The  telescope,  carrying  the 
circle  with  it.,  is  then  moved  through  the  angle  whose  value  is 
required,  and  the  new  readings  of  the  microscopes,  which  have 
remained  stationary,  are  taken.  We  thus  obtain  four  values, 
one  from  each  microscope,  of  the  angle  measured.  Theoretically, 
these  values  should  be  identical,  and  if  they  are  not,  their  mean 
is  taken  a^  the  true  measure  of  the  angle  observed. 


THE    MICROSCOPE. 


35 


Fig.  8. 


28.  The  Reading  Microscope. — The  reading  microscope  ia 
represented  in  Fig.  8. 
The  observer,  placing 
his  eye  at  A,  sees  the 
image  of  the  divisions 
of  the  graduated  circle 
MN,  formed  at  D,  the 
common  focus  of  the 
glasses  A  and  C.  He 
will  also  see  a  scale 
of  n6tches,  nn,  and 
two  intersecting  spider 
threads,  as  shown  in 
Fig.  9.  These  threads 
are  attached  to  a  sliding 
frame,  aa,  which  is 
moved  by  means  of  a 
fine  screw,  ce,  the  head 
of  which,  EFy  is  gra- 
duated. The  scale  of 
notches  is  immovable, 
and  is  so  constructed 
that  the  distance  be- 
tween the  centres  of  any 
two  consecutive  notches 
is  equal  to  that  between 
the  threads  of  the  screw,  thus  making  the  number  of  teeth  passed 
over  by  the  spider  threads  equal  to  the  number  of  complete  re- 
volutions made  by  the  screw.  The  central  notch  is  taken  as  the 
point  of  reference,  and  is  distinguished  by  a  hole  opposite  to  it. 
There  is  a  fixed  index  at  i,  to  which  the  divisions  on  the  head  of 
the  screw  are  referred.  When  any  division  of  the  limb  does  not 
coincide  with  the  central  notch,  the  spider  threads  are  moved 
from  the  central  notch  to  the  division,  and  the  number  of  revo- 
lutions and  fractional  parts  of  a  revolution  which  the  screw 
makes  is  noted.  If  now  we  suppose  the  value  of  each  division 
of  the  graduated  circle,  MN,  to  be  10',  and  that  ten  revolutions 
of  the  screw  suffice  to  carry  the  spider  threads  across  one  of  these 


F 

3-  - 

3  *• 

• 

:^j 

a 

' 

\       !              \^ 

\Wmmmmm\n  \ 

iiilMJP 

•  -8! 

/  \           aivw" 

:-r> 

Fig.  9. 


36  FIXED    POINTS. 

divisions,  then  will  one  revolution  of  the  screw  correspond  to  an 
arc  of  1' ;  and  if  we  further  suppose  that  the  head  of  the  screw 
is  divided  into  60  equal  parts,  then  each  division  on  the  head 
will  correspond  to  an  arc  of  V .  In  such  a  case,  the  complete 
reading  of  the  limb  is  obtained  to  the  nearest  second.  By  in- 
creasing the  power  of  the  microscope,  the  fineness  of  the  screw, 
and  the  number  of  the  graduations  on  the  screw-head,  the  read- 
ing of  the  limb  may  be  obtained  with  far  greater  precision. 

29.  Fixed  Points. — The  meridian  circle,  being  also  a  transit 
instrument,  may  be  used  as  such;  but  the  object  for  which  it  is 
specially  used  is  the  measurement  of  arcs  of  the  meridian.     In 
order  to  facilitate  such  measurement,  certain  fixed  points  of  re- 
ference are  determined  upon  the  vertical  circle.     The  most  im- 
portant of  these  points  are  the  horizontal  point,  by  which  is 
meant  the  reading  of  the  instrument  when  the  axis  of  the  tele- 
scope lies  in  the  plane  of  the  horizon ;  the  polar  point,  which  is 
the  reading  of  the  instrument  when  the  telescope  is  directed  to 
the  elevated  pole ;  the  zenith  point,  and  the  nadir  point. 

30.  The  Horizontal  Point. — As  the  surface  of  a  fluid,  when  at 
rest,  is  necessarily  horizontal,  and  as,  by  the  laws  of  Optics,  the 
angles  of  incidence  and  reflection  are  equal  to  each  other,  the 
image  of  a  star  reflected  in  a  basin  of  mercury  will  be  depressed 
below  the  horizon  by  an  angle  equal  to  the  altitude  of  the  star 
at  that  instant.     If,  then,  we  take  the  reading  of  the  vertical 
circle  when  a  star  which  is  about  to  cross  the  meridian  is  on  the 
first  vertical  thread  of  the  reticule,  and  then,  depressing  the 
telescope,  take  the  reading  of  the  circle  when  the  reflected  image 
of  the  star  crosses  the  last  vertical  thread,  and,  by  means  of 
small  corrections,  reduce  these  readings  to  what  they  would  have 
been,  had    both   star   and   image  been  on   the  meridian  when 
the  observations  were  made,  the  mean  of  these   two  reduced 
readings  will   be   the   horizontal  point.     The  horizontal  point 
having  been  thus  determined,  the  zenith  point  and  the  nadir  point, 
being  situated  at  intervals  of  90°  from  it,  are  at  once  obtained. 
Knowing  the  horizontal  and  the  zenith  point,  we  are  able  to 
measure  the  meridian  altitude  or  the  meridian  zenith  distance 
of  any  celestial  body  which  comes  above  our  horizon.     And 
further,  as  the  latitude  is  equal  to  the  altitude  of  the  elevated 


NADIR    POINT. 


37 


Fig.  10. 


pole,  if  tlie  latitude  of  the  place  is  accurately  known,  we  can  at 
once  obtain  the  polar  point  by  applying  the  latitude  to  the 
horizontal  point. 

31.  Nadir  Point. — The   nadir   point   may  be  independently 
obtained  in  the  following  manner.     Let  the 

telescope,  represented  in  Fig.  10  by  AJ3,  be 
directed  vertically  downwards  towards  a  basin 
of  mercury,  CD.  The  observer,  placing  his 
eye  at  A,  will  see  the  cross-wires  of  the  tele- 
scope, and  will  see  also  the  image  of  these 
•wires  reflected  into  the  telescope  from  the 
mercury.  By  slowly  moving  the  telescope, 
the  cross-wires  and  their  reflected  image  may 
be  brought  into  exact  coincidence,  and  the 
reflected  image  will  then  disappear.  The 
line  of  sight  of  the  telescope  is  now  vertical, 
and  the  reading  of  the  vertical  circle'  will  be 
the  nadir  point,  from  which  the  other  points  can  readily  be  found. 
There  is  a  variety  of  methods  by  which  each  of  these  points 
can  be  obtained,  without  reference  to  any  other;  and  by  com- 
paring the  results  which  these  different  independent  methods 
give,  the  errors  to  which  each  result  is  liable  may  be  very  con- 
siderably diminished. 

32.  Use  of  the  Meridian  Circle. — The  meridian  circle  may  be 
used  in  connection  with  the  sidereal  clock,  to  find  the  right 
ascension  and  declination  of  any  celestial  body.     The  telescope 
is  directed  towards  the  body  as  it  crosses  the  meridian,  and  the 
time  of  transit  as  shown  by  the  clock,  and  the  reading  of  the 
vertical  circle,  are  both  taken.     We  have  already  seen  how  the 
right  ascension  of  the  body  is  obtained  from  the  clock  time  of 
transit.     The  difference  between  the  reading  of  the  circle,  which 
we  suppose  to  have  been  taken,  and  the  polar  point,  is  the  polar 
distance  of  the  body,  the  complement  of  which  is  the  declina- 
tion.    Or  we  may  obtain  the  declination  still  more  directly  by 
previously  establishing  the  equinoctial  point  of  the  instrument, 
the  reading,  that  is  to  say,  of  the  vertical  circle  when  the  tele- 
scope lies  in  the  plane  of  the  equinoctial.* 

*  As  the  direction  in  which  a  star  appears  to  lie  is  not,  owing  to  refruc- 
4 


38  MURAL    CIRCLE. 

On  the  other  hand,  if  we  ma0ke  the  same  observations  upon  a 
Btar  whose  right  ascension  and  declination  are  known,  we  can 
determine  the  latitude  and  the  sidereal  time  of  the  place  of 
observation. 


THE   MURAL   CIRCLE. 

33.  The  mural  circle  is,  in  construction,  adjustment,  and  use, 
essentially  a  meridian  circle.     The  only  important  difference 
between  the  two  instruments  is  in  the  manner  in  which  they  are 
mounted.     The  horizontal  axis  of  the  mural  circle,  instead  of 
being  supported  at  both  extremities,  is  supported  only  at  one, 
which  is  let  into  a  stone  pier  or  wall.     Owing  to  the  lack  of 
symmetrical  support,  and  also  to  the  fact  that  the  instrument 
does  not  admit  of  reversal  (which  is  an  important  element  in  the 
adjustment  of  the  meridian  circle,  and  consists  in  lifting  it  out 
of  the  Ys,  and  turning  the  horizontal  axis  end  for  end),  the 
mural  circle  can  be  regarded  only  as  an  inferior  type  of  the 
meridian  circle. 

THE    ALTITUDE    AND    AZIMUTH    INSTRUMENT. 

34.  The  general  principles  on  which  the  altitude  and  azimuth 

instrument  is  constructed  are  seen 
in  Fig.  11.  Through  the  centre  of 
a  graduated  circle,  C'6",  and  per- 
pendicular to  its  plane,  is  passed  an 
axis,  A  A.  At  right  angles  to  this 
axis  is  a  second  axis,  one  extremity 
of  which  is  represented  by  B.  This 
second  axis  carries  the  telescope,  TT, 
and  also  a  second  graduated  circle, 
(7(7,  whose  plane  is  perpendicular  to 
that  of  the  circle  C"  C' '.  The  tele- 
scope admits  of  being  moved  in  the  plane  of  each  circle,  and 

tion  and  other  causes,  which  will  be  explained  in  Chap.  III.,  the  direction 
in  which  it  really  lies,  certain  small  corrections  must  be  applied  to  the 
reading  of  the  circle  to  obtain  the  reading  which  really  corresponds  to  the 
direction  of  the  star. 


ALTITUDE    AND    AZIMUTH    INSTRUMENT.  39 

microscopes  or  verniers  are  attached  to  the  instrument,  by  means 
of  which  arcs  on  either  circle  can  be  read. 

If  this  instrument  is  so  placed  that  the  principal  axis,  AA, 
lies  in  a  vertical  direction,  we  shall  have  an  altitude  and  azimuth 
instrument,  sometimes  called  an  altazimuth.  The  circle  C'Cf 
will  then  lie  in  the  plane  of  the  horizon,  and  the  axis  AA,  in- 
definitely prolonged,  will  meet  the  surface  of  the  celestial  sphere 
in  the  zenith  and  the  nadir.  The  circle  (7(7,  as  it  is  moved 
with  the  telescope  about  the  axis  AA}  will  continually  lie  in  a 
vertical  plane. 

35.  Fixed  Points. — Altitudes  maybe  measured  on  the  vertical 
circle,  when  we  know  the  horizontal  or  the  zenith  point,  the  de- 
termination of  which  has  already  been  described  in  Arts.  30  and 
31.     In  order  to  measure  the  azimuth  of  any  celestial  body,  we 
must,  in  like  manner,  establish  some  fixed  point  of  reference  on 
the  horizontal  circle,  as,  for  instance,  the  north   or  the  south 
point,  by  which  is  meant  the  reading  of  the  horizontal  circle 
when  the  telescope  lies  in  the  plane  of  the  meridian. 

36.  Method  of  Equal  Altitudes. — One  of  the  most  accurate 
methods  of  obtaining  the  north  or  the 

south  point  of  the  horizontal  circle  is 
called  the  method  of  equal  altitudes. 
Let  Fig.  12  represent  the  projection  of 
the  celestial  sphere  on  the  plane  of  the 
celestial  horizon,  NESW.  Z  is  the 
projection  of  the  zenith,  P  of  the  pole, 
and  the  arc  AA'  the  projection  of  a 
portion  of  the  diurnal  circle  of  affixed 
star,  which  is  supposed  to  have  the 
same  altitude  when  it  reaches  A',  west 
of  the  meridian,  which  it  had  at  A,  east  of  the  meridian. 

Now,  in  the  two  triangles  PAZ,  PA'Z,  we  have  PZ  common, 
PA'  equal  to  PA  (since  the  polar  distance  of  a  fixed  star  re- 
mains constant),  and  ZA  equal  to  ZA',  by  hypothesis ;  the  two 
triangles  are  therefore  equal  in  all  their  parts,  and  hence  the 
angles  PZA  and  PZA'  are  equal.  But  these' two  angles  are  the 
azimuths  of  the  star  at  the  two  positions  A  and  Af.  We  may 


40  EQUATORIAL. 

say,  then,  in  general,  that  equal,  altitudes  of  a  fixed  star  corre- 
spond to  equal  angular  distances  from  the  meridian. 

Now,  let  the  telescope  be  directed  to  some  fixed  star  east  of  the 
meridian,  and  let  the  reading  of  the  horizontal  circle  be  taken. 
When  the  star  is  at  the  same  altitude,  west  of  the  meridian,  let 
the  reading  of  the  horizontal  circle  again  be  taken ;  the  mean  of 
these  two  readings  is  the  reading  of  the  horizontal  circle  when 
the  axis  of  the  telescope  lies  in  the  plane  of  the  meridian. 

37.  Use  of  the  Altitude  and  Azimuth  Instrument. — This  instru- 
ment is  chiefly  used  for  the  determination  of  the  amount  of  re- 
fraction corresponding  to  different  altitudes.  Refraction,  as  will 
be  seen  in  the  next  chapter,  displaces  every  celestial  body  in  a 
vertical  direction,  making  its  apparent  zenith  distance  less  than 
its  true  zenith  distance.  At  the  instant  of  taking  the  altitude 
of  a  celestial  body,  the  local  sidereal  time  is  noted,  from  which, 
knowing  the  right  ascension  of  the  body,  we  can  obtain  its  hour 
angle  from  the  theorem  in  Art.  9.  We  shall  then  have  in  the 
astronomical  triangle,  PZA  (Fig.  3),  the  hour  angle  ZPA,  the 
side  PA,  or  the  polar  distance  of  the  body,  and  the  side  PZ,  the 
co-latitude  of  the  place  of  observation.  We  can,  therefore,  com- 
pute the  side  ZA,  which  is  the  true  zenith  distance  of  the  body 
observed ;  and  the  difference  between  this  and  the  observed  zenith 
distance,  (corrected  for  parallax,  [Art.  55,]  and  instrumental 
errors,)  will  be  the  amount  of  refraction  for  that  zenith  distance. 

The  construction  of  the  instrument  enables  us  to  follow  a 
celestial  body  through  its  whole  course  from  rising  to  setting, 
measuring  altitudes  and  noting  the  corresponding  times  to  any 
extent  that  we  choose;  and  the  amount  of  refraction  correspond- 
ing to  each  altitude  can  afterwards  be  computed  at  our  leisure. 


THE    EQUATORIAL. 

38.  The  equatorial  is  similar  in  general  construction  to  the 
altitude  and  azimuth  instrument.  It  is,  however,  differently 
placed,  the  plane  of  the  principal  graduated  circle,  C'  C',  coin- 
ciding, not  with  tKe  plane  of  the  horizon,  but  with  that  of  the 
celestial  equator,  from  which  peculiarity  of  position  comes  the 


EQUATORIAL.  4 1 

name  of  equatorial.  The  circle  C'C',  when  thus  placed,  is  called 
the  hour  circle  of  the  instrument,  and  the  axis  A,  at  right  angles 
to  it,  is  called  the  hour  or  polar  axw.  It  is  evident  that  the  axis 
is  directed  towards  the  poles  of  the  heavens.  The  circle  CO, 
the  plane  of  which  is  perpendicular  to  the  plane  of  the  circle 
C"  C't  will  lie  continually  in  the  plane  of  a  circle  of  declination, 
as  the  instrument  is  turned  about  the  polar  axis,  and  is  hence 
called  the  declination  circle.  The  axis  on  which  this  latter  circle 
is  mounted  is  called  the  declination  axis. 

39.    Use  of  the  Equatorial. — The  equatorial  is  employed  prin- 
cipally in  that  class  of  observations  which  require  a  celestial 
body  to  remain  in  the  field  of  view  during  a  considerable  length 
of  time.  The  manner  in  which  this 
requirement  is  met  is  explained 
in  Fig.  13.    Let  A  A'  be  the  polar 
axis  of  an  equatorial,  directed  to- 
wards the  pole  of  the  heavens,  P. 
Let  ss*s"  be  the  diurnal  circle  in 
which  a  star   appears   to    move 
about  the  pole.    Suppose  the  tele- 
scope,  TT,  to  be  turned  in  the  di- 
rection  of  the  star  when  at  «,  and  Fig.  is. 

to  be  moved  until  the  intersection  of  the  cross-wires  and  the  star 
coincide,  and  then  clamped.  Now,  if  the  instrument  is  made  to 
revolve  about  the  axis  AA't  with  an  angular  velocity  equal  to 
that  of  the  star  about  the  pole,  it  is  plain  that,  since  the  angle 
which  the  axis  of  the  telescope  makes  with  the  polar  axis  remains 
unchanged,  and  is  continually  equal  to  the  angular  distance  of 
the  star  from  the  pole,  the  coincidence  of  the  cross-wires  and  the 
star  will  remain  complete. 

A  clock-work  arrangement,  called  a  driving-clock,  is  now  usually 
connected  with  large  equafHrials,  by  which  the  instrument  may 
be  moved  uniformly  about  its  polar  axis,  at  the  required  rate,  so 
that  the  observer  has  ample  time  to  measure  the  angular  diameter 
of  a  celestial  body,  to  measure  the  angular  distance  between  two 
stars  which  are  near  each  other,  and  to  make  other  micrometric 
observations  of  a  similar  character. 


42 


SEXTANT. 


THE  SEXTANT. 

40.  The  sextant  is  an  instrument  by  which  the  angular  dis- 
tance between  two  visible  objects  may  be  measured.  It  is  used 
chiefly  by  navigators ;  but  its  portability  gives  it  great  value 
wherever  celestial  observations  are  required.  The  angles  for 
the  measurement  of  which  it  is  used  are  the  altitudes  of  celestial 
bodies,  and  the  angular  distances  between  celestial  bodies  or 
terrestrial  objects. 

Fig.  14  is  a  representation  of  the  sextant.     Its  form  is  that 


of  a  sector  of  a  circle,  the  arc  of  which  comprises  60°.  A  movable 
arm,  CD,  called  the  index-bar,  revolves  about  the  centre  of  the 
sector.  This  bar  carries  at  one  extremity  a  vernier,  D.  At  the 
other  extremity  of  the  index-bar,  and  revolving  with  it,  is  placed 
a  silvered  mirror,  C,  the  surface  of  which  must  be  perpendicular 
to  the  plane  of  the  instrument.  This  glass  is  called  the  index- 
glass.  Another  glass,  N,  called  the  horizon- glass,  is  attached  to 
the  frame  of  the  instrument,  and  only  its  lower  half  is  silvered. 


SEXTANT.  43 

This  glass  is  immovable,  and  its  surface  must  be  perpendicular 
to  the  plane  of  the  instrument.  T  is  a  telescope,  directed  to- 
wards the  horizon-glass,  with  its  line  of  sight  parallel  to  the 
plane  of  the  instrument.  F  and  E  are  two  sets  of  colored 
glasses,  which  may  be  used  to  protect  the  eye  when  the  sun  is 
observed.  M  is  a  magnifying  glass,  to  assist  the  eye  in  reading 
the  vernier.  G  is  a  tangent  screw,  which  gives  a  slight  motion  to 
the  index-bar,  and  is  used  in  obtaining  an  accurate  coincidence 
of  the  images. 

41.  Optical  Principle  of  Construction. — The   sextant   is   con- 
structed upon  a  principle  in  Optics  which  may  be  stated  thus : — 
The  angle  between  the  first  and  the  last  direction  of  a  ray  which 
has  suffered  two  reflections  in  the  same  plane  is  equal  to  twice  the 
angle  which  the  tivo  reflecting  surfaces  make  with  each  other. 

To  prove  this:  In  Fig  15, 
let  A  and  B  be  the  two  re- 
flecting surfaces,  supposed  to 
be  placed  with  their  planes 
perpendicular  to  the  plane  of 
the  paper.  Let  SA  be  a  ray 
of  light  from  some  body,  S, 

which  is  reflected  from  A  to 

B,  and  from  Bin  the  direction 
BE.  Prolong  SA  until  it  meets  Fig  15 

the  line  BE.  Then  will  the 
angle  SEB  be  the  angle  between  the  first  and  the  last  direction 
of  the  ray  SA.  At  the  points  A  and  B  let  the  lines  AD  and 
jBObe  drawn  perpendicular  to  the  reflecting  surfaces,  and  pro- 
long AD  until  it  meets  BC.  The  angle  DCB  is  equal  to  the 
angle  which  the  two  surfaces  make  with  each  other.  We  have 
then  to  prove  that  the  angle  SEB  is  double  the  angle  DCB. 

Now  since  the  angle  of  incidence  always  equals  the  angle  of 
reflection,  SAD  and  DAB  are  equal,  and  so  are  ABC  and  CBE. 
We  have,  by  Geometry, 

SEB  =  SAB  —  ABE 

=  2  (DAB  —  ABC), 

=  2  DCB. 

42.  Measurement  of  Angular  Distances. — Suppose,  now,  that 


44 


SEXTANT. 


Fig.  16. 


we  wish  to  measure  the  angular*  distance  between  two  celestial 
bodies,  A  and  B  (Fig.  16).  The  instrument  is  so  held  that  its 
plane  passes  through  the  two  bodies,  and  the  fainter  of  them, 

which  in  this  case  we  suppose  to 
be  B,  is  seen  directly  through  the 
horizon-glass   and    the    telescope. 
B  is  so  distant  that  the  rays  B'  C 
and  Bm,  coming  from  it,  may  be 
considered  to  be  sensibly  parallel. 
Let  ab  and  CI  be  the  positions  of 
the  index-glass  and  index-bar  when 
~D  the  index-glass   and   the  horizon- 
glass  are  parallel.     Then  will  the 
ray  B'C  be  reflected  by  the  two 
glasses  in  a  direction   parallel  to 
itself,  and  the  observer,  whose  eye  is 
at  D,  will  see  both  the  direct  and  the 

reflected  image  of  £  in  coincidence.  Now  let  the  index-bar  be 
moved  to  some  new  position,  CI',  so  that  the  ray  from  the  second 
body,  A,  shall  be  finally  reflected  in  the  direction  of  ml).  The 
observer  will  then  see  the  direct  image  of  B  and  the  reflected 
image  of  A  in  coincidence;  and  the  angular  distance  between 
the  two,  bodies  is  evidently  equal  to  the  angle  between  the  first 
and  the  last  direction  of  the  ray  A  C,  which  angle  has  already 
been  shown  to  be  equal  to  twice  the  angle  which  the  two  glasses 
now  make  with  each  other,  or  to  twice  the  angle  ICI'.  If,  then, 
we  know  the  point  Jon  the  graduated  arc  at  which  the  index-bar 
stands  when  the  glasses  are  parallel,  twice  the  difference  between 
the  reading  of  that  point  and  that  of  the  point  /'  will  be  the 
angular  distance  of  the  two  bodies. 

To  avoid  this  doubling  of  the  angle,  every  half  degree  of  the 
arc  is  marked  as  a  whole  degree,  when  the  graduation  is  made ; 
so  that,  in  practice,  we  have  only  to  subtract  the  reading  of  / 
from  that  of  /'  to  obtain  the  angle  required. 

43.  Index  Correction. — The  point  of  reference  on  the  arc  from 
which  all  angles  are  to  be  reckoned  is,  as  we  have  already  seen, 
the  reading  of  the  sextant  when  the  surfaces  of  the  index-glass 
and  the  horizon-glass  are  parallel.  This  point  may  fall  either 


ARTIFICIAL   HORIZON.  45 

at  the  zero  of  the  graduation,  or  to  the  left  or  to  the  right  of  it; 
and  to  provide  for  the  last  case,  the  graduation  is  carried  a  short 
distance  to  the  right  of  the  zero,  this  portion  of  the  arc  being 
called  the  extra  arc.  The  reading  of  this  point  of  parallelism 
is  called  the  index  correction,  and  is  positive  when  it  falls  to  the 
right  of  the  zero,  and  negative  when  it  falls  to  the  left.  Suppose, 
for  instance,  that  the  instrument  reads  2'  on  the  extra  arc  when 
the  glasses  are  parallel :  all  angles  ought  then  to  be  reckoned 
from  the  point  2',  instead  of  from  the  zero  point;  in  other  words, 
2'  is  a  constant  correction  to  be  added  to  every  reading. 

There  are  several  methods  of  finding  the  index  correction. 
One  method,  which  can  readily  be  shown  from  Fig.  16  to  be  a 
legitimate  one,  is  to  move  the  index-bar  until  the  direct  and  the 
reflected  image  of  the  same  star  are  in  coincidence,  and  then 
take  the  reading,  giving  it  Its  proper  sign  according  to  the  rule 
above  stated.  Another  method,  generally  more  convenient,  in 
which  the  sun  is  used,  may  be  found  in  Bowditch's  Navigator, 
and  in  most  treatises  on  Astronomy :  where  also  may  be  found 
the  methods  of  testing  the  adjustments  of  the  sextant. 

44.  The  Artificial  Horizon. — In  order  to  obtain  the  altitude 
of  a  celestial  body  at  sea,  the  sextant  is  held  in  a  vertical  posi- 
tion, and  the  index-bar  is  moved  until  the  reflected  image  of  the 
body  is  brought  into  contact  with  the  visible  horizon  seen  through 
the  telescope  of  the  sextant.  The  sextant  reading  is  then  cor- 
rected for  the  index  correction;  arid  corrections  must  also  be 
applied  for  parallax,  refraction,  and  the  dip  of  the  horizon,  as 
will  be  explained  in  the  next  Chapter.  If  the  body  observed  is 
the  sun  or  the  moon,  either  its  upper  or  its  lower  limb  is  brought 
into  contact  with  the  horizon,  and  the  value  of  its  angular 
semi-diameter  (given  in  the  Nautical  Almanac)  is  subtracted  or 
added. 

On  shore,  use  is  made  of  the  artificial  horizon,  already  alluded 
to  in  Art.  30.  This  commonly  consists  of  a  shallow,  rectangular 
basin  of  mercury,  the  surface  of  which  is  protected  from  the 
wind  by  a  sloping  roof  of  glass.  The  observer  so  places  himself 
that  he  can  see  the  image  of  the  body  whose  altitude  he  wishes 
to  measure  reflected  in  the  mercury.  He  then  moves  the  index- 
bar  of  the  sextant  until  the  image  of  the  body  reflected  by  the 


46 


VERNIER. 


sextant  is  in  coincidence  with  Jhat  reflected  by  the  mercury. 
.The  sextant  reading  is  then  corrected  for  the  whole  of  the  index 
correction.  Half  of  the  result  will  be,  as  shown  in  Art.  30, 
the  apparent  altitude  of  the  body,  to  which  must  be  applied 
the  corrections  for  parallax  and  refraction  to  obtain  the  true 
altitude.  When  the  sun  or  the  moon  is  observed,  the  upper  or 
the  lower  limb  of  the  image  reflected  by  the  sextant  is  brought 
into  contact  with  the  opposite  limb  of  the  image  reflected  by  the 
mercury,  and  the  correction  for  semi-diameter  also  is  applied. 

45.   The  Vernier. ---The  vernier  is  an  instrument  by  which,  as 
by  the  reading  microscope  previously  explained,  fractions  of  a 

division  of  a  limb  may  be  read. 
In  Fig.  17,  let  AB  be  an  arc  of 
a  stationary  graduated  circle, 
and  let  CD  be  a  movable  arm, 
carrying  another  graduated 
arc  at  its  extremity.  The  value 
of  each  division  of  the  limb 
A B  is  one-sixth  of  a  degree,  or 
10'.  The  arc  on  the  arm  CD 
is  divided  into  ten  equal  parts, 
and  the  length  of  the  arc  be- 
tween the  points  0  and  10  is 
Fi£- 17-  equal  to  the  length  of  nine  di- 

visions of  the  arc  AB.  This  arc,  which  the  lirnb  CD  carries,  is 
called  a  vernier.  Since  the  ten  divisions  of  the  vernier  equal  in 
length  nine  divisions  of  the  limb,  it  follows  that  each  division 
of  the  vernier  comprises  9'  of  arc ;  in  other  words,  any  division 
of  the  vernier  is  less  by  V  of  arc  than  any  division  of  the  limb. 
The  reading  of  any  instrument  which  carries  a  vernier  is  al- 
ways determined  by  the  position  of  the  zero  point  of  the  vernier. 
If,  now,  the  zero  point  of  the  vernier  exactly  coincides  with  a 
division  of  the  limb,  the  point  1  of  the  vernier  will  fall  V  behind 
the  next  division  of  the  limb,  the  point  2  will  fall  2'  behind  the 
next  division  but  one,  and  so  on;  and  if,  such  being  the  case, the 
vernier  is  moved  forward  through  an  arc  of  1',  the  point  1  will 
corne^nto  coincidence  with  a  division  of  the  limb;  if  it  is  moved 
forward  through  an  arc  of  2',  the  point  2  will  come  into  coiiici- 


VERNIER.  47 

dence  with  a  division  on  the  limb  ;  and,  in  general,  the  number 
of  minutes  of  -arc  by  which  the  zero  point  of  the  vernier  falls 
beyond  the  division  of  the  limb  which  immediately  precedes  it 
will  be  equal  to  the  number  of  that  point  of  the  vernier  which 
is  in  coincidence  with  a  division  of  the  limb.  If,  then,  the  zero 
point  falls  between  any  two  divisions  of  the  limb,  as  11°  20'  and 
11°  30',  for  example,  and  the  point  2  of  the  vernier  is  found  to 
be  in  coincidence  with  any  division  of  the  limb,  we  know  that 
the  zero  point  is  2'  beyond  the  division  11°  20',  and  that  the  com- 
plete reading  for  that  position  of  the  vernier  is  11°  22'. 

46.   General  Rules  of  Construction.  —  In  the  construction  of  all 
verniers  similar  to  the  one  above  described,  the  same  rules  of 
construction  must  be  followed  :  the  length  of  the  arc  of  the  ver- 
nier must  be  exactly  equal  to  the  length  of  a  certain  number 
(no  matter  what)  of  the  divisions  of  the  limb,  and  the  arc  must 
be  divided  into  equal  parts,  the  number  of  which  shall  be  greater 
by  one  than  the  number  of  these  divisions  of  the  limb.    Following 
these  rules,  and  putting 
D  —  the  value  of  a  division  of  the  limb, 
d  ±±     "         "         "         "         "          vernier, 
n  =  the  number  of  equal  parts  into  which  the  vernier  is  divided, 

we  have  D  —  d  =  —      as  a  general  formula. 

The  difference  D  —  d  is  called  the  least  count  of  the  vernier. 

If,  in  Fig.  17,  we  take  the  length  of  the  vernier  equal  to  59 
divisions  of  the  limb,  and  divide  it  into  60  equal  parts,  we  shall 
have 


which  is  the  least  count  on  most  of  the  modern  sextants. 

Verniers  are  sometimes  constructed  in  which  the  number  of 
equal  parts  on  the  vernier  is  less  by  one  than  the  number  of  the 

D 
divisions  of  the  limb  taken.     In  this  case  we  have  d  —  D  =  ~  —  • 

')!/ 

and  the  only  difference  between  this  class  of  verniers  and  the 
class  above  described  is  that  the  graduations  of  the  limb  and 
the  vernier  proceed  in  this  clas&  in  opposite  directions. 


48  SPECTROSCOPE. 

OTHER    ASTRONOMICAL    INSTRUMENTS. 

47.  The  zenith  telescope,  the  theodolite,  and  the  universal  instrn' 
ment  are,  in  general  principle,  only  modified  forms  of  the  port- 
able altitude  and  azimuth  instrument. 

The  octant  (sometimes  improperly  called  the  quadrant}  is 
identical  in  construction  with  the  sextant,  excepting  only  that 
its  arc  contains  45°. 

The  prismatic  sextant  carries  a  reflecting  prism  in  place  of  the 
ordinary  horizon-glass,  and  the  graduated  arc  comprises  a  semi- 
circumference. 

The  reflecting  circle  is  still  another  modification  of  the  sex- 
tant, in  which  the  graduated  arc  is  an  entire  circumference,  and 
the  index-bar  is  a  diameter  of  the  circle,  revolving  about  the 
centre,  and  carrying  a  vernier  at  each  extremity.  Sometimes 
the  circle  has  three  verniers,  at  intervals  of  120°  of  the  gradu- 
ated arc. 

The  spectroscope  is  an  instrument  which  is  used,  as  its  name 
indicates,  -in  the  examination  of  the  spectra  both  of  terrestrial 
substances  and  of  the  heavenly  bodies.  Its  use  as  an  instrument 
of  astronomical  research  is  comparatively  recent,  but  it  has 
already  led  to  many  interesting  and  remarkable  discoveries  con- 
cerning the  constitution  of  the  heavenly  bodies.  It  consists 
essentially  of  three  parts:  a  tube,  a  prism  (or  a  set  of  prisms), 
and  a  telescope.  Rays  of  light  from  either  a  celestial  body  or 
an  artificial  flame  are  made  to  enter  the  tube  through  an  ex- 
tremely narrow  slit  at  its  extremity.  These  rays  pass  through 
the  tube,  and  fall  upon  the  prism.  If  necessary,  lenses  may  be 
placed  within  the  tube,  so  that  the  rays,  as  they  issue  from  it, 
shall  fall  upon  the  prism  in  parallel  lines.  These  rays  are  dis- 
persed by  the  prism,  and  a  spectrum  is  formed.  This  spectrum 
is  then  examined  by  means  of  the  telescope.  There  is  also  an 
arrangement  by  .which  rays  of  light  from  two  substances  or 
bodies  can  be  introduced  through  the  slit  without  interfering 
with  each  other,  so  that  their  spectra  can  be  formed  simul- 
taneously, one  above  the  other,  and  the  points  of  resemblance  or 
difference  between  them  can  be  accurately  noted. 

It  is  well  known  that  the  solar  spectrum  contains  a  large 


KRRORS.  49 

number  of  dark  and  narrow  parallel  lines,  which  are  called 
Fraunhofer's  lines.  The  spectra  of  the  stars  and  of  artificial 
lights  also  contain  similar  series  of  lines,  differing  from  each 
other,  each  series,  however,  being  constant  for  the  same  body  or 
light.  The  spectra  of  chemical  substances  also  present  certain 
peculiarities,  so  that  each  spectrum  indicates  with  certainty  the 
substance  which  produces  it.  Hence,  by  a  comparison  of  the 
spectra  of  the  heavenly  bodies  with  those  of  known  chemical 
substances,  the  existence  of  many  of  those  substances  in  the 
heavenly  bodies  has  been  definitely  established.  Nor  is  this  all ; 
the  inspection  of  any  spectrum  suffices  to  tell  us  whether  the 
light  which  forms  it  comes  from  a  solid  or  a  gaseous  body,  and 
whether,  if  the  light  comes  from  a  solid  body,  it  passes  through 
a  gaseous  body  before  it  reaches  us. 

The  results  of  these  investigations  will  be  noticed  when  we 
come  to  the  description  of  the  heavenly  bodies ;  and  the  method 
of  investigation  will  be  further  illustrated  in  the  Article  on  the 
constitution  of  the  sun.  (Art.  102.) 


ERRORS. 

48.  However  carefully  an  instrument  may  be  constructed,  how- 
ever accurately  adjusted,  and  however  expert  the  observer  may 
be,  every  observation  must  still  be  regarded  as  subject  to  errors. 
These  errors  may  be  divided  into  two  classes,  regular  ami  -irre- 
gular errors.  By  regular  errors  we  mean  errors  which  remain 
the  same  under  the  same  combination  of  circumstances,  and 
which,  therefore,  follow  some  determinate  law,  which  may  be 
made  the  subject  of  investigation.  Among  the  most  important 
of  this  class  of  errors  are  instrumental  errors:  errors,  that  is  to 
say,  due  to  some  defect  in  the  construction  or  adjustment  of  an 
instrument.  If,  for  instance,  what  we  call  the  vertical  circle  of 
the  meridian  circle  is  not  rigorously  a  circle,  or  is  imperfectly 
graduated;  or  if  the  horizontal  axis  is  not  exactly  horizontal, 
or  does  not  lie  precisely  east  and  west ;  any  one  of  these  imper- 
fections will  affect  the  accuracy  of  the  observation.  The  observer, 
however,  knowing  what  the  construction  and  adjustment  of  the 
instrument  ought  to  be,  can  calculate  what  effect  any  given  im- 
.5 


50  ERRORS. 

perfection  will  produce  upon  his  observation,  and  can  thus  de- 
termine what  the  observation  would  have  been  had  the  imper- 
fection not  existed.  Regular  errors,  then,  may  be  neutralized 
by  determining  and  applying  the  proper  corrections. 

Irregular  errors,  on  the  contrary,  are  errors  which  are  not 
subject  to  any  known  law.  Such,  for  example,  are  errors  pro- 
duced in  the  amount  of  refraction  by  anomalous  conditions  of 
the  atmosphere ;  errors  produced  by  the  anomalous  contraction 
or  expansion  of  certain  parts  of  the  instrument,  or  by  an  un- 
steadiness of  the  telescope  produced  by  the  wind ;  and,  more  par- 
ticularly, errors  arising  from  some  imperfection  in  the  eye  or  the 
touch  of  the  observer.  Errors  such  as  these,  being  governed 
by  no  known  law,  can  never  be  made  the  subject  of  theoretic 
investigation;  but  being  by  their  very  nature  accidental,  the 
effects  which  they  produce  will  sometimes  lie  in  one  direction 
and  sometimes  in  another ;  and  hence  the  observer,  by  repeating 
his  observations,  by  changing  the  circumstances  under  which  he 
makes  them,  by  avoiding  unfavorable  conditions,  and  finally  by 
taking  the  mean,  or  the  most  probable  value  of  the  results  which 
his  different  observations  give  him,  can  very  much  diminish  the 
errors  to  which  any  single  observation  would  be  exposed. 

NOTE. — For  complete  descriptions  of  the  various  astronomical  instru- 
ments, the  student  is  referred  to  Chauvenet's  Spherical  and  Practical  As 
tronomy;  Loomis's  Practical  Astronomy;  and  Pearson's  Practical  Astronomy 
(published  in  England). 


REFRACTION.  51 


CHAPTER  III. 

REFRACTION.      PARALLAX.      DIP   OF   THE   HORIZON. 
REFRACTION. 

49.  When  a  ray  of  light  -passes  obliquely  from  one  medium 
to  another  of  different  density,  it  is  bent,  or  refracted,  from  its 
course.  If  a  line  is  drawn  perpendicular  to  the  surface  of  the 
second  medium  at  the  point  where  the  ray  meets  it,  the  ray  is 
bent  towards  this  perpendicular  if  the  second  medium  is  the 
denser  of  the  two,  and  from  it  if  the  first  medium  is  the 
denser. 

In  Fig.  18,  let  A  A,  BB,  represent  two  NEC 

media  of  different  density,  the  density  of 
BB  being  the  greater.  Let  CD  be  a 
ray  of  light  meeting  the  surface  of  BB 
at  D.  At  D  erect  the  line  ND  perpendi-  / 


cular  to  the  surface  of  BB,  and  prolong  <?    ^ 

it  in  the  direction  DM.     The  ray  CD  is  Fis- 18- 

called  the  incident  ray,  and  the  angle  ND  C  the  angle  of  inci- 
dence. When  the  ray  enters  the  medium  BB,  it  will  still  lie  in 
the  same  plane  with  CD  and  ND,  but  will  be  bent  towards  the 
line  DM,  making  with  it  some  angle  GDM,  less  than  the  angle 
ND  C.  To  an  observer  whose  eye  is  at  G,  the  ray  will  appear 
to  have  come  in  the  direction  EG,  which  is  therefore  called  the 
apparent  direction  of  the  ray.  DG  is  called  the  refracted  ray, 
and  the  angle  GDM  the  angle  of  refraction.  The  angle  EDC, 
the  difference  between  the  directions  of  the  incident  and  the 
refracted  ray,  is  called  the  refraction. 

.  It  is  shown  in  Optics  that,  whatever  the  angle  of  incidence 
may  be,  there  always  exists  a  constant  ratio  between  the  sine 
of  the  angle  of  incidence  and  that  of  the  angle  of  refraction, 
as  long  as  the  same  two  media  are  used  and  their  densities  are 
unchanged.  We  have,  then,  in  the  figure, — 


REFRACTION. 


sin  NDQ  __ 
sin  GDM=    ' 

k  being  a  constant  for  the  two  media  A  A  and  BB. 

If  the  second  medium,  instead  of  being  of  uniform  density,  is 
composed  of  parallel  strata,  each  one  of  which  is  of  greater 
density  than  the  one  immediately  preceding,  as  is  represented 
c  _  in  Fig.  19,  the  path  of  the  ray  through 
these  several  strata  will  be  a  broken 
line,  Dabc;  and  if  the  thickness  of 
~B  each  of  these  successive  strata  is  sup- 
posed to  be  indefinitely  small,  this 
broken  line  will  become  a  curve. 


In  the  figures  above  used,  the  media  are  represented  as  sepa- 
rated by  plane  surfaces;  but  the  same  phenomena  are  noticed, 
and  the  same  laws  hold  good,  if  the  media  are  separated  by 
curved  surfaces. 

50.  Astronomical  Refraction. — It  is  determined  by  experiment 
that  the  density  of  the  air  gradually  diminishes  as  we  ascend 
above  the  surface  of  the  earth,  and  it  is  estimated  that  at  a 
distance  of  fifty  miles  above  the  surface  the  upper  limit  of  the 
air  is  reached ;  or,  at  all  events,  that  the  density  of  the  air  is  so 
small  at  that  distance  that  it  exerts  no  appreciable  refracting 
power.  We  may,  therefore,  consider  the  air  to  be  made  up  of 
a  series  of  strata  concentric  with  the  earth's  surface,  the  thick- 
ness of  each  stratum  being  in- 
definitely small,  and  the  den- 
sity of  each  stratum  being 
greater  than  that  of  the  stra- 
tum next  above  it.  Now,  in 
s  Fig.  20,  let  the  arc  BD  repre- 
sent a  portion  of  the  earth's 
surface,  and  the  arc  MN  a  por- 
tion of  the  upper  limit  of  the 
atmosphere.  Let  8  be  a  ce- 
lestial body,  and  SA  a  ray  of 
light  from  it,  which  enters  the 
atmosphere  at  A.  Let  the 
normal  (or  radius)  AC  be 


REFRACTION  53 

drawn.  As  the  ray  of  light  passes  down  through  the  atmos- 
phere, it  is  continually  passing  from  a  rarer  to  a  denser 
medium,  so  that  its  path  is  continually  changed,  and  becomes 
a  curve  AL,  concave  towards  the  earth,  and  reaching  the  earth 
at  some  point  L.  Since  the  direction  of  a  curve  at  any  point 
is  the  direction  of  the  tangent  to  the  curve  at  that  point,  the 
apparent  direction  of  the  ray  of  light  at  L  will  be  repre- 
sented by  the  tangent  LSf,  and  in  that  direction  will  the  body 
S  appear  to  lie,  to  an  observer  at  L.  If  the  radius  OL'be 
indefinitely  prolonged,  the  point  Z,  where  it  reaches  the  celestial 
sphere,  will  be  the  zenith  of  the  observer  at  L,  the  angle  ZLS' 
will  be  the  apparent  zenith  distance  of  the  body  8,  and  the 
angle  which  the  line  drawn  from  the  body  to  the  point  L 
makes  with  the  line  LZ  will  be  the  true  zenith  distance  of  S. 
The  effect,  then,  of  refraction  is  to  decrease  the  apparent  zenith 
distances,  or  increase  the  apparent  altitudes  of  the  celestial 
bodies.  Since  the  incident  ray  SA,  the  curve  AL,  and  the 
tangent  LS'  all  lie  in  the  same  vertical  plane,  the  azimuth  of  the 
celestial  bodies  is  not  affected. 

51.  General  Laws  of  Refraction. — By  an  investigation  of  the 
formulae  of  refraction,  and  by  astronomical  observations  already 
described  (Art.  37),  the  amount  of  refraction  at  different  alti- 
tudes has  been  obtained,  and  is  given  in  what  are  called  "tables 
of  refraction."  The  following  general  laws  of  refraction  will 
serve  to  give  the  student  some  idea  of  its  amount,  and  of  the 
conditions  under  which  it  varies: — 

(1.)  In  the  zenith  there  is  no  refraction. 

(2.)  The  refraction  is  at  its  maximum  in  the  horizon,  being 
there  equal  to  about  33'.  At  an  altitude  of  45°  it  amounts 
to  57". 

(3.)  For  zenith  distances  which  are  not  very  large,  the  re- 
fraction is  nearly  proportional  to  the  tangent  of  the  zenith 
distance.  When  the  zenith  distance  is  large,  however,  the  ex- 
pression of  the  law  is  much  more  complicated.  No  table  of  re- 
fraction can  be  trusted  for  an  altitude  of  less  than  5°. 

(4.)  The  amount  of  refraction  depends  upon  the  density  of  the 
air,  and  is  nearly  proportional  to  it.  The  tables  give  the  re- 
fraction for  a  mean  state  of  the  atmosphere,  taken  with  the 


54  PARALLAX. 

barometer  at  30  inches  and  the  thermometer  at  50°.  If  the 
temperature  remains  constant,  and  the  barometer  stands  above 
its  mean  height,  or  if  the  height  of  the  barometer  is  constant, 
and  the  thermometer  stands  below  its  mean  height,  the  density 
:)f  the  atmosphere  is  increased,  and  the  refraction  is  greater 
than  its  mean  amount.  Supplementary  tables  are  therefore 
given,  from  which,  with  the  observed  heights  of  both  barometer 
and  thermometer  as  arguments,  we  may  take  the  necessary  cor- 
rections to  be  applied  to  the  mean  refraction. 

(5.)  Since  the  effect  of  refraction  is  to  increase  the  apparent 
altitudes  of  the  celestial  bodies,  the  amount  of  refraction  for 
any  apparent  altitude  is  to  be  subtracted  from  that  apparent 
altitude,  or  added  to  the  corresponding  zenith  distance. 

52.  Effects  of  Refraction. — The  apparent  angular  diameter  of 
the  sun  and  of  the  moon  being  about  32',  and  the  refraction  iu 
the  horizon  being  33',  it  follows  that  when  the  lower  limb  of 
either  body  appears  to  be  resting  on  the  horizon,  the  body  is  m 
reality  below  it.     One  effect,  then,  of  refraction  is  to  lengthen 
the  time  during  which  these  bodies  are  visible.     Still  another 
effect  is  to  distort  the  discs  of  the  sun  and  the  moon  when  near 
the  horizon :    for  since  the  refraction  varies  rapidly  near  the 
horizon,  the  lower  extremity  of  the  vertical  diameter  of  the 
body  will  be  more  raised  than  the  upper  extremity,  thus  appa- 
rently shortening  this  diameter,  and  giving  the  body  an  ellip- 
tical shape.     When  the  body  comes  still  nearer  the  horizon,  its 
disc  is  distorted  into  what  is  neither  a  circle  nor  an  elljpse,  but 
a  species  of  oval,  in  which  the  curvature  of  the  lower  limb  is 
less  than  that  of  the  upper  one.     The  apparent  enlargement  of 
these  bodies  when  near  the  horizon  is  merely  an  optical  delu- 
sion, which  vanishes  when  their  diameters  are  measured  with  an 
ii.strument. 

PARALLAX. 

53.  The  parallax  of  any  object  is,  in  the  general  sense  of  the 
word,  the  difference  of  the  directions  of  the  straight  lines  drawn 
to  that  object  from  two  different  points:    or  it  is  the  angle  at 
the  object  subtended  by  the  straight  line  connecting  these  two 
points.     In  Astronomy,  we  consider  two  kinds  of  parallax :  yco- 


PARALLAX. 


55 


centric  parallax,  by  which  is  meant  the  difference  of  the  directions 

of  the  straight  lines  drawn  to  the  centre  of  any  celestial  body  from 

the  earth's  centre  and  any  point  on  its  surface,  and  heliocentric 

parallax,  or  the  difference  of  the  directions  of  the  lines  drawn  to 

the  centre  of  the  body  from  the  cen- 

tre of  the  earth  and  the  centre  of  the 

sun.     The  former  is  the  angle  at  the 

body  subtended  by  that   radius  of 

the  earth  which  passes  through  the  sf 

place  of  observation  :  the  latter  the 

angle  at  the  body  subtended  by  the 

straight  line  joining  the  centre  of 

the  earth  and  that  of  the  sun. 

54.  Geocentric  Parallax.  —  In  Fig. 
21,  let  Obe  the  centre  of  the  earth, 
and  L  some  point  on  its  surface,  of  which  Z  is  the  zenith.  Let 
S-  be  some  celestial  body.  The  geocentric  parallax  of  the  body 
is  the  angle  CSL.  Let  S>  be  the  same  body  in  the  horizon. 
The  angle  LS'  C  is  the  parallax  of  the  body  for  that  position, 
and  is  called  its  horizontal  parallax.  If  we  denote  this  horizontal 
parallax  by  P,  the  earth's  radius  by  E,  and  the  distance  of  the 
body  from  the  earth's  centre  by  d,  we  have,  by  Trigonometry, 


Fig.  21. 


To  find  the  parallax  for  any  other  position,  as  at  S,  we  repre- 
sent the  angle  LSC  by  p,  and  the  apparent  zenith  distance 
of  the  body,  or  the  angle  ZLS,  by  z,  the  sine  of  which  is  equal 
to  the  sine  of  its  supplement  SLC.  We  have  from  the  tri- 
angle LSC,  since  the  sides  of  a  plane  triangle  are  proportional 
to  the  sines  of  their  opposite  angles, 

sin  p       R 

sin  z        d 
Combining  this  equation  with  the  preceding,  we  have, 

sin  p  =  sin  Psin  z. 

Since  P  and  p  are  small  angles,  we  may  consider  them  propor- 
tional to  their  sines,  and  thus  have,  finally, 

p  —  P  sin  z. 
The  parallax,  then,  is   proportional   to  the  sine  of  the  zenith 


56  PARALLAX. 

distance,  and  may  be  found  for  any  altitude  when  the  hori 
zontal  parallax  is  known.  It  evidently  decreases  as  the  alti- 
tude increases,  and  in  the  zenith  becomes  zero. 

55.  Application   of   Parallax.  —  In    order   that   observations 
made  at  different  points  of  the  earth's  surface  may  be  com- 
pared, they  must  be  reduced  to  some  common  point.     Geocen- 
tric parallax  is  applied  to  reduce  any  altitude  observed  at  any 
place  to  what  it  would  have  been  had  it  been  observed  at  the 
earth's  centre.     We  see  from  Fig.  21  that  parallax  acts  in  a 
vertical  plane,  and  that  the  zenith  distance  of  the  body  as  ob- 
served from  the  earth's  centre,  or  the  angle  ZC8,  is  less  than  the 
observed  zenith  distance  ZLS,  by  the  parallax  CSL.     Parallax, 
then,  is  always  subtractive  from  the  observed  zenith  distance, 
and  additive  to  the  observed  altitude. 

The  parallax  above  described  is,  strictly  speaking,  the  paral- 
lax in  altitude.  There  is  also,  in  general,  a  similar  parallax  in 
right  ascension,  and  in  declination,  formulae  for  deriving  which 
from  the  parallax  in  altitude  are  given  in  other  works. 

56.  Heliocentric  Parallax. — It  may  sometimes  happen  that  we 
wish  to  reduce  an  observation  from  what  it  was  at  the  centre 
of  the  earth  to  what  it  would  have  been  if  it  had  been  made 
at  the  centre  of  the  sun.     Fig.  21,  and  the   formulae   obtained 
from  it,  will  apply  equally  well  to  this  case,  by  making  the 
necessary  changes  in  the  description  of  the  figure  and  in  the 
names  of  the  angles. 

Let  S  be  still  a  celestial  body,  but  let  C  be  the  centre  of  the 
sun,  and  L  that  of  the  earth.  The  angle  p  will  then  represent 
the  heliocentric  parallax,  and  the  angle  SLCihe  angular  dis- 
tance of  the  body  from  the  sun,  as  measured  from  the  earth's 
centre,  or,  as  it  is  called,  the  body's  elongation.  The  angle  P 
will  be  the  greatest  value  of  the  heliocentric  parallax,  taken 
when  the  body's  elongation  from  the  sun  is  90°,  and  is  called 
the  annual  parallax.  We  shall  then  find,  from  the  formulae 
of  Art.  54,  that  the  annual  parallax  has  for  its  sine  the  ratio 
of  the  distance  of  the  earth  from  the  sun  to  that  of  the  body 
from  the  sun,  and  that  the  parallax  for  any  other  position  is 
the  product  of  the  annual  parallax  by  the  sine  of  the  body's 
elongation. 


DIP   OF   THE   HOEIZON. 


57 


Fig.  22. 


DIP    OF   THE    HORIZON. 

57.  The  dip  of  the  horizon  is  the  angular  depression  of  the 
visible  horizon  below  the  celestial  hori-  B 
zon.  In  Fig.  22,  let  HG  be  a  por- 
tion of  the  earth's  surface,  and  C 
the  earth's  centre.  Let  a  radius  of 
the  earth,  CA,  be  prolonged  to  some 
point  D,  beyond  the  surface,  and  let 
an  observer  be  supposed  to  be  at  the 
point  D.  At  the  point  D  let  the  line 
BD  be  drawn  perpendicular  to  the  line 
CD,  and  also  the  line  DH,  tangent  to 
the  earth's  surface  at  some  point  H.  If  these  two  lines  be  revolved 
about  the  line  CD,  DB  will  generate  the  plane  of  the  celestial 
horizon  (since  we  have  seen  that  all  planes  passed  perpendicular 
to  the  radius  will,  when  indefinitely  extended,  mark  out  the 
same  great  circle  on  the  celestial  sphere),  and  DH  will  gene- 
rate the  surface  of  a  cone,  which  will  touch  the  earth  in  a 
small  circle.  If  we  disregard  for  the  present  the  effect  of  the 
earth's  atmosphere,  this  small  circle  will  be  the  visible  horizon 
of  the  observer  at  D,  and  the  angle  BDHvfill  be  the  dip.  DA 
is  the  linear  height  of  the  observer  at  D. 

Now  let  a  radius,  CH,  be  drawn  to  the  point  of  tangency  H. 
The  angles  BDH  and  HCD,  having  their  sides  mutually  per- 
pendicular, are  equal.  Represent  the  angle  HCD  by  D,  the 
earth's  radius  by  J2,  and  the  observer's  height,  AD,  by  h.  ID. 
the  triangle  DHC,  right-angled  at  H,  we  have, 


since  D  is  small,  it  is  better  to  use  the  formula, 

cos  D  =  1  —  2  sin8  *  D  : 
hence  we  have, 

=1  —  28infiD: 


58 


DIP   OF   THE    HORIZON. 


As  the  angle  D  is  small,  we  mav  take 

sin  \  D  =  J  D  sin  1"  : 

and  as  h  is  very  small  in  comparison  with  JR,  we  may  also 
assume  (E -{- h~)  to  be  sensibly  equal  to  E.  Making  these 
changes  in  the  above  equation,  and  finding  the  value  of  D,  we 
have 


Substituting  in  this  expression  the  value  of  the  earth's  radius 
in  feet,  we  have,  finally, 

D  =  63".8  y~h^ 
h  being  expressed  in  feet. 

The  dip,  then,  for  a  height  of  one  foot  is  63".8 ;  and  for 
other  heights  it  is  proportional  to  the  square  root  of  the  number 
of  feet  in  the  height. 

58.  Effect  of  Atmospheric  Refraction. 
— If  the  effect  of  atmospheric  refrac- 
tion is  taken  into  consideration,  the 
G  line  HD  must  be  a  curved  line,  as  is 
represented  in  Fig.  23.    The  point  H 
will  then  appear  to  lie  in  the   direc- 
tion DHf,  to  the  observer  at  D,  and 
the  dip  will  be  the  angle  BDH '.    The 
Fis- 23-*  effect,  then,  of  refraction  is  to  decrease 

the  dip,  the  amount  by  which  it  is  decreased  being  about  -j-^th 
of  the  whole.  • 

59.  Application  of  Dip.  —  Tables  have  been  computed  in 
which  may  be  found  the  proper  dip  for  different  heights  above 
the  surface  of  the  earth.  Dip  constitutes  one  of  the  correc- 
tions which  are  to  be  applied  at  sea  to  the  observed  altitude  of 
a  celestial  body  to  obtain  its  true  altitude:  its  altitude,  that  is, 
above  the  celestial  horizon.  Since  the  visible  horizon  lies  below 
the  celestial  horizon,  this  correction  is  evidently  subtractive. 


The  curved  line  HD  is  tangent  to  the  earth's  surface  at  H. 


MEASUREMENT   OF   THE   EARTH.  .r>9 


CHAPTER  IV. 

THE   EARTH.      ITS   SIZE,    FORM,   AND   ROTATION. 

60.  HAVING  seen  what  are  the  construction  and  the  adjustments 
of  the  principal  astronomical  instruments,  to  what  uses  each  is 
adapted,  and  what  corrections  are  to  be  applied  to  the  observa- 
tions which  are  taken,  we  are  now  ready  to  proceed  to  the  solu- 
tion of  some  of  the  many  questions  of  interest  which  the  study 
of  Astronomy  opens  to  us.     And  first  of  all,  let  us  see  how  as- 
tronomical observations  will  help  us  to  a  knowledge  of  the  size 
and  the  form  of  the  earth.     The  question  is  one  of  the  first  im- 
portance; for  upon  the  determination  of  the  size  of  the  earth 
depends  in  a  great  measure,  as  we  shall  see  further  on,  the  deter- 
mination of  the  magnitudes   and   the   distances  of  the   other 
heavenly  bodies.     Having  seen  the  facts  which  seem  to  point  to 
the  conclusion  that  the  earth  is  spherical  in  form,  wre  will  start 
with  the  assumption  that  this  conclusion  is  correct,  and  proceed 
to  determine  the  magnitude  of  the  earth,  regarded  as  a  sphere. 

Now,  we  know  from  Geometry  what  is  the  ratio  between  the 
radius  of  a  sphere  and  the  circumference  of  any  great  circle  of 
that  sphere;  and,  therefore,  if  we  can  obtain  the  length  of  the 
circumference  of  any  great  circle  of  the  earth,  of  a  meridian, 
for  instance,  we  can  at  once  determine  the  radius  of  the  earth. 
And  more  than  this :  if  we  can  measure  the  length  of  any  known 
arc  of  this  meridian,  of  one  degree,  for  instance,  we  can  compute 
the  length  of  the  entire  circumference.  The  determination  of 
the  earth's  radius,  then,  depends  only  on  our  ability  to  satisfy 
these  two  conditions: 

(1.)  We  must  be  able  to  measure  the  linear  distance  on  the 
earth's  surface  between  two  points  on  the  same  meridian. 

(2.)  We  must  be  able  to  measure  the  angular  distance  between 
these  same  two  points. 

61.  First  Condition. — A  reference  to  Fig.  24  will  explain  how 


00 


MEASUREMENT    OF   THE    EARTH. 


Fig.  24. 


the  first  of  these  two  conditions  may  be  satisfied.  Let  A  and  G 
represent  the  two  points  on  the  same  meridian  the 
distance  between  which  we  wish  to  measure.  We 
have  already  seen  (Art.  36)  how  an  altitude  and 
azimuth  instrument  may  be  so  adjusted  that  the 
sight-line  of  the  telescope  will  lie  in  the  plane  of 
the  meridian.  Let  an  instrument  be  so  adjusted 
at  the  point  A.  Let  some  convenient  station  B  be 
taken,  visible  at  A,  and  let  the  distance  AB  be 
carefully  measured.  This  distance  is  called  the 
base-line.  Now,  by  means  of  the  telescope,  adjusted 
to  the  plane  of  the  meridian,  let  some  point  C  be 
established  on  the  meridian,  which  shall  also  be 
visible  from  B,  and  let  the  angles  CAB  and  ABC 
be  measured.  We  now  know  in  the  triangle  ABC 
two  angles  and  the  included  side,  and  can  compute  the  distances 
AC  and  CB.  We  now  take  some  suitable  point  D,  measure 
the  angles  DCB  and  DBC,  and  knowing  CB,  we  can  obtain  the 
distance  CD.  The  instrument  is  now  taken  to  the  point  C,  and 
again  established  in  the  plane  of  the  meridian,  either  by  the 
method  of  Art.  36,  or  by  sighting  back,  as  it  is  called,  to  A,  or  by 
both  methods  combined.  A  third  point  on  the  meridian,  E, 
visible  from  D,  is  then  selected,  the  angles  ECD  and  ED  C  are 
measured,  and  the  distances  EC  and  ED  computed.  This  pro- 
cess is  continued  until  the  whole  distance  between  A  and  G  has 
been  obtained. 

This  method  of  measurement  is  called  the  method  of  triangu- 
lation.  The  base-line,  AB,  is  purposely  taken  under  circum- 
stances which  favor  its  accurate  measurement,  and  the  rest  of 
the  work  consists  in  the  determination  of  horizontal  angles, 
which  presents  no  special  difficulty,  and  in  the  solution  of  tri- 
angles by  computation. 

Two  things  are  to  be  noticed  in  reference  to  these  triangles. 
The  first  is  that  in  selecting  the  points  B,  C,  D,  &c.,  care  must  be 
taken  so  to  choose  them  that  the  triangles  ABC,  BCD,  &c.,  shall 
be  nearly  equiangular;  since  triangles  in  which  there  is  a  great 
inequality  of  the  angles  (ill-conditioned  triangles,  as  they  are 
called)  will  be  much  more  likely  to  cause  some  error  in  the  work. 


MEASUREMENT    OF   THE    EARTH.  61 

The  second  thing  to  be  noticed  is  that  these  triangles  are  really 
spherical  triangles,  and  must  therefore  be  solved  by  the  formulae 
of  Spherical  Trigonometry.  If,  for  any  reason,  we  are  forced 
to  take  any  of  the  points  C,  E,  &c.,  off  the  meridian,  the  cor- 
responding distances  can  be  reduced  to  the  meridian  by  appro- 
priate formulae. 

The  correctness  of  the  result  may  be  tested  by  measuring  the 
distance  GF,  and  comparing  its  measured  length  with  that  ob- 
tained by  computation.  The  marvellous  accuracy  of  this  method 
of  measurement  is  shown  by  the  fact  that  in  an  arc  of  the  meri- 
dian measured  by  the  French  at  the  close  of  the  last  century, 
and  which  was  several  hundred  miles  in  length,  the  discrepancy 
between  the  measured  and  the  computed  length  of  the  second 
base-line  was  less  than  twelve  inches. 

62.  Second  Condition. — The  second  condition  requires  that 
the  angle  at  the  centre  of  the  earth,  subtended  by  the  arc  of  the 
meridian  measured,  shall  be  obtained.  This  angle  is  evidently 
the  difference  of  latitude  of  the  two  extremities  of  the  arc,  and 
therefore  all  that  is  needed  to  satisfy  this  condition  is  that  the 
latitude  of  each  extremity  shall  be  determined  by  appropriate 
observations. 

Instead  of  determining  the  latitude  of  each  place  independently 
of  the  other,  we  may,  if  we  choose,  obtain  the  difference  of 
latitude  directly,  by  observing  at  each  place  the  meridian  zenith 
distance  of  the  same  celestial  body.  In  Fig.  25,  let  A  and  G  be 
the  two  extremities  of  the  arc,  C  the  centre 
of  the  earth,  and  S  the  celestial  body  on 
the  meridian.  If  Z'  is  the  zenith  of  the 
point  A,  the  meridian  zenith  distance  of  8 
at  A,  reduced  to  the  centre  of  the  earth,  is 
the  angle  Z'  CS.  In  the  same  manner  the 
true  meridian  zenith  distance  of  S  at  G  is 
the  angle  ZCS.  The  difference  of  these  two 
zenith  distances,  or  the  angle  ZCZ\  is  evi- 
dently the  difference  of  latitude  of  G  and  A.  Fis- 25- 

If  the  celestial  body  crosses  the  meridian  between  the  l\vo 
zeniths,  as  at  $',  the  difference  of  latitude  is  numerically  the 
sum  of  the  two  meridian  zenith  distances. 


62  FORM    OF    THE    EARTH. 

63.  Results. — By  the  process  ajbove  described,  or  by  processes 
of  a  similar  character,  arcs  of  different  meridians,  and  in  differ- 
ent latitudes,  have  been  carefully  measured.     The  sum  of  the 
arcs  thus  measured  is  more  than  60°,  and  the  length  of  a  degree 
of  the  meridian  has  been  found  to  be,  on  the  average,  69.05  miles. 
Multiplying  this  by  360,  we  obtain  24,858  miles  for  the  circum- 
ference of  a  meridian,  and  dividing  this  circumference  by  -, 
(3.1416)  we  find  the  length  of  the  earth's  diameter  to  be  7912 
miles. 

64.  Spheroidal  Form  of  the  Earth. — One  remarkable  fact  is 
noticed  when  we  compare  the  lengths  of  the  degrees  of  the  meri- 
dian, measured  in  different  latitudes ;  and  that  is,  that  the  length 
of  the  degree  is  not  the  same  at  all  parts  of  the  meridian,  but  sen- 
sibly increases  as  we  leave  the  equator.     The  length  of  a  degree 
at  the  equator  is  found  to  be  68.7  miles,  whilst  at  the  poles  it  is 
computed  to  be  69.4  miles.     The  conclusion  drawn  from  this 
fact  is  that  the  figure  of  the  earth  is  not  rigorously  that  of  a 
sphere,  since  a  spherical  form  necessarily  implies  an  absolute 
uniformity  in  the  length  of  a  degree  in  all  parts  of  a  great  circle. 
In  order  to  determine  the  exact  geometrical  figure  of  the  earth, 
we  must  bear  in  mind  that  the  curvature  of  a  line  is  always  pro- 
portional to  the  change  in  the  direction  of  the  tangents  drawn 
at  successive  points  of  that  line.     Now,  since  the  altitude  of  the 
elevated  pole  at  any  place  is  equal  to  the  latitude  of  that  place, 
it  follows  that  an  advance  towards  the  pole  of  one  degree  in  lati- 
tude is  accompanied  by  a  depression  of  one  degree  in  the  plane  of 
the  horizon.     If,  therefore,  in  order  to  effect  a  change  of  one  de- 
gree in  our  latitude,  we  are  forced  to  advance  a  greater  number 
of  miles  at  the  pole  than  at  the  equator,  we  conclude  that  the  cur- 
vature of  the  meridian  is  less  at  the  pole  than  at  the  equator.     Now, 

this  same  inequality  in  its  curvature  is 
also  a  peculiarity  of  the  ellipse :  and  hence 
we  infer  that  the  form  of  the  earth's  me- 
ridians is  not  that  of  a  circle,  but  that 
of  an  ellipse,  as  represented  in  Fig.  26. 
The  axis  of  the  earth,  Pp,  corresponds 
to  the  minor  axis  of  an  ellipse,  at  the 
extremities  of  which  the  curvature  is  the 


DENSITY.  63 

least;  and  the  equatorial  diameter  of  the  earth,  EQ,  corresponds 
to  the  major  axis  of  an  ellipse,  at  the  extremities  of  which  the 
curvature  is  the  greatest. 

The  form  of  the  earth,  then,  is  that  of  the  solid  which  would 
be  generated  by  the  revolution  of  an  ellipse  about  its  minor 
axis,  which  solid  is  called  in  Geometry  an  oblate  spheroid.  A 
more  common  but  less  accurate  name  given  to  the  form  of  the 
earth  is  that  of  a  sphere,  flattened  at  the  poles. 

65.  Dimensions  of  the  Earth.  —  The  following  are  the  dimen- 
sions of  the  earth,  when  its  spheroidal  form  is  taken  into  con- 
sideration.    The  determination  is  that  of    Sir  G.  B.  Airy,  the 
Astronomer  Royal  of  England. 

Polar  diameter  .........................  7899.170  miles. 

Equatorial  diameter  ...................  7925.648  miles. 

These  values  are  believed  to  be  within  a  quarter  of  a  mile  of  the 
true  values.  They  differ  from  the  results  obtained  by  the  astro- 
nomer Bessel  by  only  about  o^th  of  a  mile. 

The  compression,  or  oblateness,  of  an  oblate  spheroid  is  the 
ratio  of  the  difference  between  the  major  and  the  minor  axis 
of  the  generating  ellipse  to  its  major  axis.  The  compression  of 

the  earth  is  therefore          '    .    :  which  is  about        th. 


If  a  and  b  represent  the  semi-major  and  the  semi-minor  axis 
of  the  generating  ellipse,  the  expression  for  the  volume  of  the 
oblate  spheroid  is  $xa'2b.  Substituting  in  this  expression  the 
values  of  a  and  b,  we  find  the  earth's  volume  to  be  about  260 
billions  of  cubic  miles. 

66.  Density  of  the  Earth.  —  There  are  various  methods  of  de- 
termining the  mean  density  of  the  earth.  The  following  is  a 
brief  summary  of  the  method  of  determining  it  by  means  of  the 
torsion  balance.  This  balance  consists  of  a  slender  wooden  rod, 
supported  in  a  horizontal  position  by  a  very  fine  wire  at  its 
centre.  To  the  extremities  of  this  rod  are  attached  two  small 
leaden  balls.  If  left  free  to  move,  this  horizontal  rod  will  of 
course  come  to  rest  when  the  supporting  wire  is  free  from 
rorsion.  Two  much  larger  leaden  balls  are  now  brought  near 
the  two  suspended  balls,  and  on  opposite  sides,  so  that  the 
attractions  of  both  balls  may  combine  to  twist  the  wire  in 


64  ROTATION. 

same  direction.  The  smaller  balls  will  be  sensibly  attracted 
by  the  larger  ones,  and  the  horizontal  rod  will  change  the 
direction  in  which  it  lies.  The  amount  of  this  deflection  is 
very  carefully  measured,  and  from  it  is  computed  the  attraction 
which  the  large  balls  exert  on  the  small  ones.  But  we  know 
the  attraction  which  the  earth  exerts  on  the  small  balls,  it  being 
represented  by  their  weight:  and  we  know  also  the  volumes 
of  the  earth  and  the  attracting  balls.  Finally,  we  know  the 
density  of  lead:  and  from  these  data  it  is  possible  to  compute 
the  mean  density  of  the  earth. 

A  series  of  over  2000  experiments  of  this  nature  was  con- 
ducted in  England,  in  1842,  by  Sir  Francis  Baily.  The  mean 
density  of  the  earth,  obtained  from  these  experiments,  was  5.67 : 
the  density  of  water  being  the  unit.  Other  methods  of  deter- 
mining the  density  of  the  earth  have  been  employed,  the  main 
principle  in  each  method  being  the  comparison  of  the  at- 
traction exerted  by  the  earth  upon  some  object  with  that  ex- 
erted by  some  other  body,  whose  density  can  be  ascertained, 
upon  the  same  object.  The  results  of  these  experiments  do  not 
differ  materially  from  the  results  of  the  experiments  with  the 
torsion  balance. 

The  volume  and  density  of  the  earth  being  known,  what  is 
commonly  called  its  weight  can  be  computed.  It  is  found  to 
be  about  six  sextillions  of  tons. 

ROTATION  OF  THE  EARTH. 

67.  Up  to  this  point  we  have  assumed  the  earth  to  be  at 
rest,  and  the  apparent  diurnal  motions  of  the  heavenly  bodies 
to  be  real  motions.      By  careful  observation  of  the  sun,  the 
moon,   and  the  most  conspicuous  of  the  planets,   astronomers 
have  demonstrated  that  each  of  these  bodies  rotates  upon  a 
fixed  axis.     Analogy,  therefore,  points  to  a  similar  rotation  of 
our  own  planet:  and  besides  this,  there  are  many  phenomena 
which  are  inexplicable  if  the  earth  is  at  rest,  but  which  are 
fully  accounted  for  on  the  supposition  that  it  rotates  upon  an 
axis.    We  will  now  examine  the  principal  of  these  phenomena. 

68.  The  weight  of  the  same  body  is  not  the  same  in  different 
latitudes.     Careful  experiments  made  in  different  latitudes  show 


CENTRIFUGAL    FORCE.  65 

that  the  weight  of  the  same  body  is  not  constant  at  all  parts 
of  the  earth's  surface,  but  increases  with  the  latitude.  A  body 
which  weighs  194  pounds  at  the  equator  will  weigh  195  pounds 
if  taken  to  either  pole  ;  that  is  to  say,  the  weight  of  any  body  is 
increased  by  j^th  of  itself  when  carried  from  the  equator  to 
the  pole.  This  experiment  cannot  be  made  with  the  ordinary 
balances  in  which  bodies  are  weighed  :  since  it  is  obvious  that 
the  same  cause,  whatever  it  may  be,  which  affects  the  weight 
of  the  body  will  also  affect  that  of  the  weights  by  which  it  is 
balanced,  and  by  the  same  amount,  so  that  the  scales  will  still 
remain  in  equilibrium.  If,  however,  we  test  the  weight  of  a 
body  (the  force,  that  is  to  say,  with  which  it  tends  to  the  earth's 
centre)  by  the  effect  which  it  has  in  stretching  a  spring,  the 
increase  of  weight  will  be  found  to  be  as  stated  above. 

Part  of  this  increase  of  weight  is  due  to  the  spheroidal  form 
of  the  earth,  since  a  body  when  at  the  pole  is  nearer  the  centre 
of  the  earth  than  when  at  the  equator.  The  amount  of  increase 
due  to  this  cause  has  been  calculated  to  be  about  ^ih;  hence 
the  difference  between  j^th  and  g^o^nJ  which  is  2^-gth,  still  re- 
mains to  be  accounted  for.  We  shall  now  see  how  it  is  com- 
pletely accounted  for  by  the  supposition  that  it  is  the  effect  of 
the  centrifugal  force  which  is  induced  by  a  rotation  of  the  earth 
upon  its  polar  axis. 

69.  Centrifugal  force.  —  The  tendency  which  a  body  has,  when 
revolving  about  any  point  as  a  centre,  to  recede  from  that  centre, 
is  called  its  centrifugal  force.  The  formula  for  the  centrifugal 
force  may  be  found  in  any  treatise  on  Mechanics,  and  is  as 
follows  : 


in  which  /  is  the  centrifugal  force,  r  the 
radius  of  the  circle  of  revolution,  and  t  the 
veriodic  time,  or  the  time  in  which  the 
revolution  is  performed.  Now,  in  Fig.  27 
let  the  earth  be  supposed  to  rotate  about 
its  polar  axis,  Pp,  once  in  every  sidereal 
day,  which,  as  we  have  already  seen 
(Art.  7),  is  3m.  56s.  less  than  the  mean 


66  CENTRIFUGAL    FOIICE. 

solar  day,  and  therefore  contains  86,164s.  Substituting  thi.« 
value  of  t  in  the  formula  given  al>ove,  and  substituting  for  r  the. 
value  of  the  earth's  equatorial  radius  in  feet,  and  computing 
the  value  of/,  we  shall  find  that  the  centrifugal  force  at  the 
equator  is  .1113  feet.  Now  the  actual  force  of  gravity  at  the 
equator  is  found,  by  Mechanics,  to  be  32.09  feet.  If  the  earth 
were  at  rest,  the  force  of  gravity  at  the  equator  would  evidently 
be  32.09  +  -1H3  feet.  Hence  the  diminution  of  gravity  at 
the  equator,  due  to  centrifugal  force,  (in  other  words,  the  loss 
of  weight),  is  equal  to  y^f^,  or  5|^th. 

Since  the  periodic  time  (t  in  the  formula)  is  constant  for  all 
places  on  the  earth's  surface,  it  is  evident,  from  the  formula,  that 
the  centrifugal  force  at  any  place  L  is  to  the  centrifugal  force 
at  the  equator  as  the  radius  of  revolution  at  L,  or  LM,  is  to  CQ 
But  we  have  in  the  figure, 

ML        ML 

-QQ~  =  ~QL  —  cos  «<& C  —  cos  Latitude. 

Denoting,  then,  the  centrifugal  force  at  the  equator  by  C, 
and  that  at  L  by  c,  we  have, — 

c  =  C  cos  L: 
or  the  centrifugal  force  varies  with  the  cosine  of  the  latitude. 

The  centrifugal  force  at  L  acts  in  the  direction  of  the  radius 
of  revolution  ML.  Let  its  amount  be  represented  by  LB, 
taken  on  LM  prolonged.  This  force  may  be  resolved  into  two 
forces  :  LA,  in  the  direction  from  the  centre  of  the  earth,  and  AB, 
at  right  angles  to  LA.  The  force  LA,  being  directly  opposed 
to  the  attraction  of  the  earth,  has  the  effect  of  diminishing  the 
weight  of  bodies  at  L,  and  may  therefore  be  taken  to  represent 
the  loss  of  weight  at  L. 

Denoting  the  loss  of  weight  by  w,  and  the  centrifugal  force 
at  L  by  c,  as  before,  we  have,  from  the  triangle  ABL, 
w  =  c  cos  L. 

But  we  have  already, — 

c  =  C  cos  L : 
/.  w  =  C  cos2  L. 

Now,  at  the  equator,  as  is  evident  from  the  figure,  the  whole 
effect  of  the  centrifugal  force  is  exerted  to  diminish  the  weight 
of  bodies,  and  C  therefore  also  represents  the  loss  of  weight  at 


TRADE    WINDS.  07 

the  equator.  We. have  then,  finally,  that  the  loss  of  weight  of  a 
body  at  any  latitude,  due  to  centrifugal  force,  is  equal  to  the  pro- 
duct of  -.j^th  of  the  weight  multiplied  by  the  square  of  the  cosine 
of  the  latitude. 

70.  Spheroidal  Form  of  the  Earth  due  to  Centrifugal  Force. — 
We  see,  then,  that  the  supposition  that  the  earth  rotates  upon  its 
axis  fully  explains  the  observed  difference  in  the  weight  of  the 
same  body  in  different  latitudes.     But  this  is  not  all :   for  if  we 
assume  that  the  particles  of  matter  of  which  the  earth  is  com- 
posed were  formerly  in  a  fluid  or  molten  condition,  and  there- 
fore free  to  move,  the  spheroidal  form  of  the  earth  is  itself  a 
proof  of  the  earth's  rotation.     Numerous  experiments  may  be 
made  to  show  that,  for  a  fluid  body  at  rest,  the  form  of  equili- 
brium is  that  of  a  sphere :   and  that,  for  a  fluid  body  which 
rotates,  the  form  of  equilibrium  is  that  of  a  spheroid,  the  oblate- 
ness  of  which  increases  with  the  velocity  of  rotation.     Knowing 
the  volume  and  the  density  of  the  earth,  and  assuming  the  time 
of  rotation  to  be  twenty-four  sidereal  hours,  it  is  possible  to 
calculate  the  form  of  equilibrium  which  a  fluid  mass  under 
these  conditions  will  assume :  and  this  form  is  found  to  be  that 
of  a  spheroid,  with  an  oblateness  very  nearly  identical  with  the 
known  oblateness  of  the  earth. 

This  tendency  of  a  fluid  mass  to  assume  a  spheroidal  form 
under  rotation  may  also  be  shown  in  Fig.  27.  The  centrifugal 
force  LB  was  resolved  into  the  two  forces  LA  and  AJB,  the 
former  of  which  forces  has  already  been  discussed.  The  effect 
of  the  latter  force,  AB,  is  evidently  a  tendency  in  the  particle 
L  to  move  towards  the  equator  EQ;  and  a  similar  force  acting 
upon  all  the  particles  of  matter  on  the  earth's  surface,  excepting 
those  at  the  poles  and  at  the  equator,  will  cause  them  all  to 
move  in  the  direction  of  the  equator,  and  thus  give  a  spheroidal 
form  to  the  mass. 

71.  Trade    Winds. — The   trade   winds   are  permanent  winds 
which  prevail  in  and  sometimes  beyond  the  torrid  zone.     These 
winds  are  northeasterly  in  the  northern  hemisphere  and  south- 
easterly in  the  southern  hemisphere.     The  air  within  the  torrid 
zone  being,  in  general,  subject  to  a  greater  degree  of  heat  than 
the  air  at  other  portions  of  the  earth's  surface,  rises,  and  its 


68  PENDULUM    EXPERIMENT. 

place  is  filled  by  air  which  comes  in  from  the  regions  beyond  the 
tropics.  If  the  earth  were  at  res*t,  these  currents  of  air  would 
manifestly  have  simply  a  northerly  and  a  southerly  direction. 
Now,  we  all  know  that,  when  we  travel  in  any  direction  on  a 
still  day,  or  even  when  the  wind  is  moving  in  the  same  direc- 
tion with  us,  but  with  a  less  velocity,  the  wind  seems  to  come 
from  the  point  towards  which  we  are  going.  We  see  from  Fig. 
27  that,  if  the  earth  is  rotating  upon  its  polar  axis,  the  linear 
velocity  of  rotation  decreases  as  the  latitude  increases.  Hence, 
the  air  from  beyond  the  tropics,  having  at  the  start  only  the 
linear  velocity  of  the  place  which  it  leaves,  will,  as  it  moves 
towards  the  equator,  have  continually  a  less  velocity  than  that 
of  the  surface  over  which  it  passes,  and  will  seem  to  come  from 
the  quarter  towards  which  those  places  are  moving.  If,  then, 
the  earth  is  rotating  from  west  to  east,  these  currents  of  air  will 
have  an  apparent  motion  from  the  east,  which  motion,  when 
compounded  with  the  motion  from  the  north  and  the  south, 
before  mentioned,  will  give  us  the  northeasterly  and  south- 
easterly winds  which  we  call  the  Trades. 

72.  The  Pendulum  Experiment* — The  last  and  decidedly  the 
most  satisfactory  proof  of  the  earth's  rotation  which  we  shall 
notice,  is  that  which  comes  from  the  apparent  rotation  of  the 
plane  of  a  freely-suspended  pendulum,  when  made  to  vibrate 
at  any  point  on  the  earth's  surface  except  the  equator. 

It  is  an  established  law  in  Mechanics  that  a  pendulum,  freely 
suspended  from  a  fixed  point,  always  vibrates  in  the  same  plane ; 
and  also  that  if  we  give  the  point  of  support  a  slow  movement 
of  rotation  about  a  vertical  axis,  the  plane  of  vibration  will  still 
remain  unchanged.  If,  for  instance,  we  suspend  a  ball  by  a 
string,  and,  having  caused  it  to  vibrate,  twist  the  string,  the  ball 
will  rotate  about  the  axis  of  the  string,  while  the  plane  in  which 
it  vibrates  will  not  be  affected. 

Now,  let  us  suppose  that  a  pendulum  is  suspended  at  the  north 
pole,  and  is  made  to  vibrate:  and  let  us  further  suppose  that  the 
earth  rotates  from  west  to  east,  once  in  24  hours.  The  line  in 
which  the  plane  of  vibration  intersects  the  plane  of  the  horizon 

*  This  is  called  FoucaulCs  experiment.  A  full  discussion  of  it  is  given  in 
the  American  Journal  of  Science,  2d  series,  vols.  XII-XIV. 


PENDULUM    EXPERIMENT.  69 

will  move  about  in  the  plane  of  the  horizon,  in  a  direction  oppo- 
site to  that  in  which  the  earth  is  rotating,  and  with  an  equal 
velocity,  thus  completing  one  revolution  in  24  hours.  In  Fig. 
28,  let  AGED  be  the  horizon  of  the  ob- 
server at  the  north  pole,  and  let  the  earth 
rotate  in  the  direction  indicated  by  the  (  \\y  \\ 

arrows.  Let  the  pendulum  at  P  be  set 
swinging  in  the  direction  of  some  diameter, 
AB,  of  the  horizon.  At  the  end  of  an  hour, 
the  rotation  of  the  earth  will  have  carried 
this  diameter  to  some  new  position  A'B', 
at  the  end  of  the  next  hour  to  some  new  position  A"J3",  &c. : 
while  the  pendulum  will  still  swing  in  the  original  direction  AB. 
To  the  observer,  then,  unconscious  of  the  earth's  rotation,  the 
plane  of  vibration,  which  really  remains  unchanged,  will  appear 
to  rotate  in  a  direction  opposite  to  that  in  which  the  earth  is 
rotating. 

At  the  south  pole,  under  the  same  suppositions,  a  similar  phe- 
nomenon will  be  noticed,  except  that  the  plane  of  vibration  will 
apparently  move  in  the  opposite  direction.  Thus,  if  at  the  north 
pole  the  apparent  motion  of  the  plane  is  like  that  of  the  hands 
of  a  clock,  as  we  look  on  its  face,  the  apparent  motion  at  tho 
south  pole  will  be  the  opposite  to  this. 

Again,  if  a  pendulum  is  made  to*  vibrate  in  the  plane  of  a 
meridian  at  the  equator,  there  will  be  no  apparent  change  in  the 
plane  of  vibration,  since  it  will  always  coincide  with  the  plane 
of  the  meridian,  and  hence  the  pendulum  will  continue  to 
swing  north  and  south  during  the  entire  period  of  the  earth's 
rotation.  The  condition  that  the  pendulum  shall  here  swing  in 
the  plane  of  a  meridian  is  entirely  unnecessary,  and  is  made 
only  for  the  sake  of  illustration ;  for  there  will  be  no  apparent 
change  in  the  plane  of  vibration,  whatever  may  be  the  direction 
in  which  the  pendulum  is  made  to  vibrate. 

The  apparent  rotation,  then,  of  the  plane  of  vibration  of  the 
pendulum  is  360°  in  24  hours  at  the  poles,  and  nothing  at  the 
equator.  At  places  lying  between  the  equator  and  the  poles, 
the  apparent  angular  motion  of  the  plane  of  vibration  will  be 
between  these  two  limits;  in  other  words,  less  than  360°  in 


70  LINEAR   VELOCITY    OF    ROTATION. 

24  hours.  Appropriate  investigations  show  that  the  apparent 
angular  motion  of  the  plane  of* vibration  at  any  place  in  any 
interval  of  time  is  equal  to  the  angular  amount  of  the  earth's 
rotation  in  that  time,  multiplied  by  the  sine  of  the  latitude  of 
the  place.*  Thus,  at  Annapolis,  we  have  for  the  angular  motion 
in  one  hour, 

15°  sin  38°  59' =  9°  26': 

so  that  the  plane  of  vibration  will  make  one  apparent  rotation 
at  Annapolis  in  38h.  09m. 

Such  is  the  theory  of  the  pendulum  experiment.  Now,  nume- 
rous experiments  have  been  made  in  different  latitudes,  and  in 
every  case  an  apparent  rotation  of  the  plane  of  vibration  from 
east  to  west  has  been  observed,  with  a  rate  agreeing  very  closely 
with  that  demanded  by  the  theory ;  and  the  conclusion  is  irre- 
sistible that  the  earth  rotates  on  its  polar  axis,  from  west  to  east, 
once  in  every  sidereal  day. 

73.  Linear  Velocity  of  Rotation. — Taking  the  equatorial  cir- 
cumference of  the  earth  to  be  24,900  miles,  we  have  a  linear 
velocity  of  over  1000  miles  an  hour,  and  over  17  miles  a  minute. 
This  is  the  velocity  at  the  equator.  The  linear  velocity  at  other 
points  on  the  earth's  surface  is  less  than  this,  since  the  circum- 
ferences of  the  parallels  of  latitude  are  less  than  the  circumfer- 
ence of  the  equator.  Since  the  circumference  of  any  parallel  is 

*  This  formula  may  be  obtained  by  the  principles  of  the  resolution  of  ro- 
tation, given  in  treatises  on  Mechanics.  Thus,  in  the 
figure,  the  rotation  of  the  point  L  about  the  axis  of 
the  earth,  PO,  may  be  resolved  into  two  rotations, 
one  about  the  radius  LO,  and  the  other  about  the 
radius  MO,  drawn  perpendicular  to  LO.  If  v  re- 
presents  the  angular  velocity  of  L  about  the  axis 
PO  (or  15°  in  one  hour),  and  vf  and  v"  the  angular  velocities  about  the 
axes  LO  and  MO,  we  have,  from  Mechanics, 

v'  =  v  cos  LOP,  and  v"  —  v  cos  POM. 

NOAV,  the  rotation  about  the  axis  OM  will  have  no  effect  in  changing  the 
apparent  position  of  the  plane  of  vibration  of  the  pendulum,  since  it  is 
analogous  to  the  case  at  the  equator  considered  in  the  text ;  while  the  rota- 
tion about  the  axis  LO,  being  analogous  to  the  case  at  the  pole,  will  pro- 
duce a  similar  effect.  The  apparent  angular  motion,  then,  of  the  plane 
of  vibration  will  be  v  cos  LOP,  or  v  sin  Lat. 


LINEAR   VELOCITY   OF    ROTATION.  71 

to  that  of  the  equator  as  the  radius  of  the  parallel  is  to  the 
radius  of  the  equator,  the  linear  velocity  will  diminish  as  we 
leave  the  equator  in  the  same  ratio  that  the  radii  of  the  succes- 
sive parallels  diminish :  in  the  ratio,  that  is,  of  the  cosine  of  the 
latitude,  as  was  proved  in  Art.  69.  For  instance,  the  cosine  of 
60°  being  J,  the  linear  volooHv  nt  thst  latitude  is  on'y  8£ 
a  minute. 


72  LATITUDE. 


CHAPTEK  V. 

LATITUDE.      LONGITUDE. 
LATITUDE. 

74.  THE  latitude  of  any  place  on  the  earth's  surface  has  been 
proved,  in  Articles  10  and  11,  to  be  equal  to  either  the  altitude 
of  the  elevated  pole  or  the  declination  of  the  zenith  at  that 
place.     We  shall  now  proceed  to  explain  the  principal  methods 
by  which  either  one  or  the  other  of  these  arcs  may  be  found. 

75.  First  Method. — Let  Fig.  29  represent  a  projection  of  the 

celestial  sphere  on  the  plane  of  the  celestial 
meridian,  RZHN,  of  some  place.  HR  is 
the  celestial  horizon  at  that  place,  Z  the 
zenith,  P  the  elevated  pole,  and  EQ  the 
equator.  Let  s  represent  some  circumpolar 
star,  whose  declination  is  known,  at  its 
lower  culmination.  Let  its  meridian  alti- 
tude be  observed,  and  corrected  for  instru- 
mental errors  and  refraction.  (For  all  celestial  bodies  except  the 
sun,  the  (moon,  and  the  planets,  the  corrections  for  parallax  and 
semi-diameter  will  be  inappreciable.)  To  this  corrected  altitude 
add  the  star's  polar  distance,  the  complement  of  the  star's  known 
declination.  The  sum  is  the  altitude  of  the  elevated  pole,  or 
the  latitude. 

If  the  circumpolar  star  is  at  its  upper  culmination,  as  at  $', 
the  polar  distance  is  to  be  subtracted  from  the  corrected  altitude. 
If  Tir  and  h  denote  the  corrected  altitudes  at  the  upper  and 
the  lower  culmination,  p'  and  p  the  corresponding  polar  dis- 
tances, and  L  the  latitude,  we  have  evidently 
L  =  h'—p' 
L  =  h  +  /; : 
whence  L  =  2  (k'  -f  //.)  -f  }  (/>  —p)- 


LATITUDE.  1 0 

In  this  formula  the  value  of  the  latitude  does  not  depend  on  the 
absolute  value  of  either  polar  distance,  but  merely  on  the  chart  ye 
of  the  polar  distance  between  the  two  transits,  which  is  usually 
s?o  small  as  to  be  neglected.  This  method,  then,  is  free  from  any 
error  in  the  declination,  and  is  used  at  all  fixed  observatories. 

76.  Second  Method. — When  the  star  is  at  its  upper  culmina- 
tion, it  will,  in  general,  be  more  convenient  to  find  the  declina- 
tion of  the  zenith  from  the  meridian  zenith  distance  of  the  star. 
Taking  the  star  s',  for  instance,  and  denoting  its  meridian  zenith 
distance  by  z,  and  its  declination  by  d,  we  have 

L  =  ZQ  =  Qs'  —  Zs'  ^d  —  z.  (a) 

For  the  star  s",  we  have 

L  =  Zs"  +  Qs"  =  z  +  d,  (b) 

and  for  the  star  s" 

L  =  Zs"  —  Qs"  =  z  —  d.  (c) 

From  these  three  formulae  a  general  rule  may  be  deduced,  appli- 
cable to  the  upper  culmination  of  every  star.  We  notice  that 
in  the  formulae  (a)  and  (6),  where  d  is  positive,  the  stars  s'  and 
s"  are  on  the  same  side  of  the  equator  with  the  elevated  pole; 
that  is  to  say,  their  declinations  have  the  same  name  as  the  ele- 
vated pole;  while  in  the  formula  (c)  the  declination  has  the 
^.opposite  name.  We  also  notice  that  in  the  formulae  (6)  and  (r,}t 
where  z  is  positive,  the  stars  are  on  the  opposite  side  of  the  zenith 
from  the  elevated  pole;  in  other  words,  their  bearing  has  the  op- 
posite name  to  that  of  the  pole :  while  the  bearing  of  the  star  «', 
in  the  formula  for  which  z  is  negative,  has  the  same  name  as  the 
elevated  pole.  The  general  rule,  then,  for  all  these  stajs  will  be 
the  following: — If  the  star  bears  south,  mark  the  zenith  distance 
lorth;  if  it  bears  north, 'mark  the  zenith  distance  south;  mark 
the  declination  north  or  south,  as  the  star  is  north  or  south  of 
vJie  equator,  and  combine  the  zenith  distance  and  the  declina- 
tion, thus  marked,  according  to  their  names. 

77.  Third  Method. — A  very  successful  adaptation  of  the  pre- 
ceding method  is  made  by  using  two  stars  which  culminate  at 
nearly  the  same  time,  but  on  opposite  sides  of  the  zenith,  as  s' 
and  s"  in  Fig.  29.     These  two  stars  are  so  selected  that  the  dif- 
ference of  theii  zenith  distances  is  very  small,  and  can  be  mea- 


74  LATITUDE. 

sured  directly  by  means  of  a  micrometer.  By  the  formulae  of 
the  preceding  article  we  have  for  a', 

L^d  —  z, 

and  for  s",  denoting  its  meridian  zenith  distance  and  declinatioii 
by  z'  and  d', 

L  =  d'  +  z', 
whence  we  have, 

L  =  J  (d  +  d")  -f  i-  (zr  —  z\ 

The  determination  of  the  latitude  is  thus  made  free  from  any 
error  in  the  graduations  of  the  vertical  circle,  and  depends  only 
on  the  known  declinations  of  the  two  stars,  and  on  the  difference 
of  their  zenith  distances.  Errors  in  the  refraction  are  also  very 
nearly  eliminated. 

This  is  the  principle  of  what  is  called  Talcott's  Method,  a 
method  very  commonly  used  by  the  United  States  Coast  Survey. 
The  instrument  employed  is  the  zenith  telescope,  a  modification 
of  the  altitude  and  azimuth  instrument.  The  two  stars  are  so 
selected  that  the  difference  of  their  zenith  distances  is  less  than 
the  breadth  of  the  field  of  the  telescope.  The  instrument  is  set 
in  the  plane  of  the  meridian  to  the  mean  of  the  two  zenith  dis- 
tances, and  for  the  star  which  culminates  first.  When  this  star 
crosses  the  meridian,  it  is  bisected  by  the  micrometer  wire,  and 
the  micrometer  is  read.  The  instrument  is  then  turned  180°  in 
azimuth,  and  the  process  is  repeated  with  the  second  star.  The 
difference  of  the  zenith  distances  is  then  obtained  from  the  dif- 
ference of  the  two  micrometer  readings,  and  added  to  the  half 
sum  of  the  two  declinations,  according  to  the  formula. 

78.  Fgurth  Method. — When  the  local  time  (either  solar  or 
sidereal)  is  known,  the  latitude  may  be  obtained  from  altitudes 
which  are  not  measured  on  the  meridian.  Let  Fig.  30  be  a  pro- 
jection of  the  celestial  sphere  on  the  plane 
of  the  horizon.  Z  is  the  zenith  of  the  place, 
P  the  elevated  pole,  PZthe  co-latitude,  and 
S  a  star,  whose  altitude  is  measured.  SPZ 
is  the  hour  angle  of  the  star,  which  can  be 
obtained  from  the  local  time  noted  at  the 
instant  the  altitude  is  observed.  PS  is  the 
star's  known  polar  distance.  In  the  triangle 


REDUCTION    OF    THE    LATITUDE. 


SPZ,  we  have  the  sides  ZS  and  SP,  and  the  angle  SPZ,  and  can 
therefore  compute  the  value  of  the  co-latitude,  PZ,  by  the  for- 
mulae of  Spherical  Trigonometry. 

An  analytical  investigation  of  the  formulae  by  which  this  pro- 
blem is  solved  shows  that  errors  in  the  observed  altitude  and  the 
time  have  the  less  effect  upon  the  result  the  nearer  the  body  is 
to  the  meridian. 

79.  Methods  of  Finding  the  Latitude  at  Sea. — The  second  and 
the  fourth  of  the  methods  above  described  are  the  methods  most 
commonly  employed  in  finding  the  latitude  at  sea.     The  sun  is 
the  body  which  is  generally  used,  its  altitude  above  the  sea 
horizon  being  measured  with  a  sexta*nt  or  an  octant.     The  time  of 
noon  being  approximately  known,  the  observer  begins  to  measure 
the  altitude  of  the  lower  limb  of  the  sun  a  few  minutes  before 
noon,  and  continues  to  measure  it  until  the  sun  ceases  to  rise,  or 
"dips,"  as  it  is  called.     The  greatest  altitude  which  the  sun 
attains  is  considered  to  be  the  meridian  altitude,  although,  rigor- 
ously speaking,  it  is  not.     The  proper  corrections  for  index-error, 
dip,  refraction,  parallax,  and  semi-diameter  are  next  applied  to 
the  sextant  reading,  and  the  result  is  the  sun's  true  meridian 
altitude,  from  which  the  latitude  is  obtained  by  the  rule  given  in 
Art.  76. 

When  cloudy  weather  prevents  the  determination  of  the  meri- 
dian altitude  of  either  the  sun  or  any  other  celestial  body,  an 
altitude  obtained  within  an  hour  of  transit,  on  either  side  of  the 
meridian,  maybe  used  to  find  the  latitude  by  the  fourth  method, 
Art.  78.  Bowditch's  Navigator  contains  special  tables  by  which 
the  computation,  particularly  when  the  sun  is  observed,  may  bo 
greatly  facilitated. 

80.  Reduction  of  the  Latitude. — 
Owing  to  the  spheroidal  form  of  the 
earth,  the  vertical  line  at  any  point 
of  the  surface,  as  Z'  0'  in  Fig.  31, 
which    corresponds     exactly    with 
the   normal  drawn    at   that   point, 
does  not  coincide  with  the  radius  of 
the  earth,  LO,  passing  through  the 

point,  excepting  at  the  equator 


76  LONGITUDE. 

and  the  poles.  It  is  necessary,  Jhen,  in  refined  observations,  to 
distinguish  between  the  geographical  zenith,  Z1 ',  the  point  where 
the  vertical  line,  when  prolonged,  meets  the  celestial  spheie,  and 
the  geocentric  zenith,  Z,  the  point  in  which  the  radius  meets  the 
sphere.  Since  there  are  two  zeniths,  there  are  also  two  lati- 
tudes :  Z'  0'  Q,  the  geographical  latitude,  and  ZOQ,  the  geocentric 
latitude.  The  geographical  latitude  is  evidently  greater  than 
the  geocentric,  by  the  angle  OLO',  which  is  called  the  reduction 
of  the  latitude.  Formulae  and  tables  for  finding  this  reduction 
are  given  in  Chauvenet's  Astronomy.  It  is  only  considered  in 
cases  where  the  highest  accuracy  in  the  results  is  required. 


LONGITUDE. 

81.  Let  Fig.  32  represent  a  projection  of  the  celestial  sphere 
on  the  plane  of  the  equinoctial  ABCG.     P  is  the  projection  of 
the  elevated  pole,  and  PG,  PA,  and  PB  are 
projections  of  arcs  of  great   circles    of   the 
sphere   passing   through   the  pole.     L?t  PG 
represent  the  projection  of  the  meridian  of 
Greenwich,  PA  that  of  the  meridian  of  some 
other  place  on  the  earth's  surface,  and  PJ> 
that    of    the    circle    of    declination    passing 
Fi«-  32-  through  some  celestial   body  S.     Then  will 

the  angle  CPA  represent  the  longitude  of  the  meridian  PA 
from  Greenwich,  GPU  will  represent  the  Greenwich  hour  angle 
of  the  body  S,  and  APB  will  represent  its  hour  angle  from  the 
meridian  PA.  The  difference  between  these  two  hour  angles  is 
evidently  equal  to  the  longitude  of  any  place  on  the  meridian 
PA.  The  longitude,  then,  of  any  place  on  the  earth's  surface 
is  equal  to  the  difference  of  the  hour  angles  of  the  same  celestial 
Lody  at  that  place  and  at  Greenwich,  at  the  same  absolute  instant 
of  time.  When  the  Greenwich  hour  angle  is  the  greater  of  these 
two  hour  angles,  reckoned  always  to  the  west,  the  longitude  of 
the  place  is  west :  when  it  is  the  smaller,  the  longitude  is  east. 

If  PB  is  the  hour  circle  passing  through  the  sun,  the  longi- 
tude of  the  place  is  the  difference  of  the  solar  times  at  the  place 
and  at  Greenwich:  if  it  is  the  hour  circle  passing  through  the 


CHRONOMETERS.  77 

vernal  cqainox,  the  longitude  is  the  difference  of  the  two  side- 
real times.  In  order,  then,  to  determine  the  longitude  of  any 
place,  we  nmst  be  able  to  determine  both  the  local  and  the 
Greenwich  i/me  (either  sidereal  or  solar)  at  the  same  instant. 

There  are-  various  methods  of  obtaining  the  local  time,  one 
of  which  hhs  already  been  described  (Art.  20).  It  may  be 
noticed  here  that  we  are  always  able,  by  means  of  the  Nautical 
Almanac,  to  convert  sidereal  time  into  solar  time,  or  solar  into 
sidereal  (Art.  105).  It  remains,  then,  to  determine  the  Green- 
wich time,  either  sidereal  or  solar,  to  do  which  several  distinct 
methods  may  6e  employed. 

82.  Greenw.ck  Time  by  Chronometers. — If  a  chronometer  is 
accurately  regulated  to  Greenwich  time,  that  is  to  say,  if  the 
amount  by  which  it  is  fast  or  slow  at  Greenwich  on  any  day, 
and  its  daily  .gain  or  loss,  are  determined  by  observation,  the 
chronometer  can  be  carried  to  any  other  place  the  longitude 
of  which  is  desired,  and  the  Greenwich  time  which  the  chrono- 
meter gives  can  be  directly  compared  with  the  time  at  that  place. 
This  would  be  a  perfectly  accurate  method,  if  the  rate  of  the 
chronometer  remained  constant  during  the  transportation;  but, 
in  fact,  the  rate  of  a  chronometer  while  it  is  carried  from  place 
to  place   is  very  rarely  exactly  the  same  that  it  is  while  the 
chronometer  is  at  rest.    By  using  several  chronometers,  however, 
and  by  transporting  them  several  times  in  both  directions  be- 
tween the  two  places,  and  finally  by  taking  a  mean  of  all  the 
results,  the  error  may  be  reduced  to  a  very  minute  amount. 
For  instance,  the   longitude  of  Cambridge,  Mass.,  was  deter- 
mined by  means  of  fifty  chronometers,  which  were  carried  three 
times  to  Liverpool  and  back,  and  from  them  the  longitude  was 
obtained  with  a  probable  error  of  only  4th  of  a  second  of  time. 

83.  Greenwich  Time  by  Celestial  Phenomena. — There  are  cer- 
tain celestial  phenomena  which  are  visible  at  the  same  absolute 
instant  of  time,  at  all  places  where  they  can  be  seen  at  all.     Such 
are  the  beginning  and  the  end  of  a  lunar  eclipse ;  the  eclipses 
of  the   satellites   of  the   planet  Jupiter  by  that   planet ;   the 
transits  of  these  satellites  across  the  planet's  disc,  and  their  nc- 
cultations  by  it.     The  Greenwich  times  at  which  these  various 
phenomena  will  occur  arc  computed  beforehand,  and  are  pub- 


78  LUNAR   DISTANCES. 

Jished  in  the  Nautical  Almanac.  .  The  observer,  then,  to  obtain 
his  longitude,  has  only  to  note  the  local  time  at  which  any  one 
of  these  phenomena  occurs,  and  to  compare  that  time  with  the 
corresponding  Greenwich  time  given  in  the  Almanac.  The  dif- 
ficulty of  determining  the  exact  instant  at  which  these  phe- 
nomena occur,  however,  diminishes  to  some  extent  the  accuracy 
of  the  results.  On  the  other  hand,  the  times  of  solar  eclipses 
mid  of  occultations  of  stars  by  the  moon,  although  not  identical 
at  different  places,  can  be  very  accurately  determined :  and 
iicnce  these  phenomena  are  often  employed  in  obtaining  longi- 
tudes. (Art.  164.) 

84.  Greenwich  Time  by  Lunar  Distances. — By  the  lunar  distance 
of  a  celestial  body  is  meant  its  true  angular  distance  from  the 
centre  of  the  moon,  as  it  would  be  seen  at  the  centre  of  the 
earth.     The  lunar  distances  of  the  sun,  of  the  four  brightest 
planets,  and  of  nine   bright  stars    are  given   in   the  Nautical 
Almanac,  computed  for  every  third   hour  of  Greenwich   solar 
time.     An  observer,  then,  who  wishes  to  determine  his  longitude, 
measures  the  apparetd  angular  distance  of  the  moon  from  some 
one  of  these  bodies,  and  also  notes  the  local  time  at  which  the 
observation  is  made.     He  then  finds  from  this  apparent  distance, 
by  means  of  appropriate  formulae  and  tables,  the  true  geocentric 
angular  distance,  at  the  time  of  observation,  between  the  two 
bodies.     He  then  enters  the  table  of  lunar  distances   in   the 
Almanac  with  this  distance,  and  finds  the  corresponding  Green- 
wich time,  from  which,  and  the  local  time  noted,  he  can  deter- 
mine his  longitude. 

85.  Difference  of  Longitude  by  Electric  Telegraph. — When  two 
stations,  the  difference  of  longitude  of  which  is  desired,   are 
connected  by  an  electro-telegraphic  wire,  the  difference  of  longi- 
tude may  be  determined  by  means  of  signals  made  at  either 
station,  and  recorded  at  both.     Suppose,  for  instance,  there  are 
two  stations  A  and  B,  of  which  A  is  the  more  easterly,  and 
that  each  station  is  provided  with  a  clock  regulated  to  its  own 
local  time.     Let  the  observer  at  A  make  a  signal,  the  time  of 
which  is  recorded  at  each  station.     Let  A  denote  the  difference 
of  longitude  of  the  two  stations,  T  the  local  time  at  A  at  which 
the  signal  is  made,  and  T'  the  corresponding  time  at  B.     Since 


ELECTRIC   SIGNALS.  79 

A  is  to  the  east  of  B,  its  time  is  greater  at  any  instant  tlinn 
that  of  B.  We  have  then,  supposing  the  signal  to  be  recorded 
simultaneously  at  the  two  stations, 

A^r— r. 

Experience  proves,  however,  that  the  records  of  the  signal 
are  not  exactly  simultaneous,  since  time  is  required  for  the 
electric  current  to  pass  over  the  wire.  In  the  example  above 
given,  then,  if  we  denote  the  time  required  by  the  electric  fluid 
to  pass  from  A  to  B  by  x,  the  time  recorded  at  B  will  evidently 
be,  not  T',  but  T'  -|-  * ;  so  that  the  expression  for  the  difference 
c»f  longitude  will  be 

l'=T—  T'—  x. 

Now  let  us  suppose  that  instead  of  the  signal's  being  made 
by  the  observer  at  .4,  it  is  made  by  the  observer  at  B,  at  the 
time  T'.  The  corresponding  time  recorded  at  A  will  not  be  I7, 
but  T  -\-  x.  In  this  case,  then,  the  expression  for  the  difference 
of  longitude  will  be 

r=  T+  x—  r. 

Taking  the  mean  of  the  values  of  A'  and  /',  we  have 
I  ()!  _|_  ;/')  =-  T—  T'=L 

Any  error,  therefore,  which  is  caused  by  the  time  consumed 
by  the  electric  current  in  passing  between  two  stations  is  elimi- 
nated by  determining  the  difference  of  longitude  by  signals  made 
at  both  stations,  and  taking  the  mean  of  the  results. 

86.  Difference  of  Longitude  by  "Star  Signals." — The  "  method 
of  star  signals"  is  a  modification  of  the  method  describe^  in 
the  preceding  paragraph,  which  is  extensively  used  in  the 
United  States  Coast  Survey.  The  principle  on  which  this 
method  rests  is  that,  since  a  fixed  star  makes  one  apparent  revo- 
lution about  the  earth  in  exactly  twenty-four  sidereal  hours, 
the  difference  of  longitude  between  two  meridians  is  equal  to 
the  interval  of  sidereal  time  in  which  any  fixed  star  passes  from 
one  of  these  meridians  to  the  other.  The  clock  by  which  this 
interval  of  time  is  measured  may  be  placed  at  either  station, 
or  indeed  at  any  place  which  is  in  telegraphic  communication 
with  both  stations.  Two  chronographs,  one  at  each  station,  are 
connected  with  the  wire  and  the  clock,  and  upon  them  are 


SO  LONGITUDE   AT   SEA. 

recorded,  by  breaks  in  the  circuit  as  explained  in  Art.  22,  the 
successive  beats  of  the  clock.  A  transit  instrument  is  adjusted 
to  the  meridian  at  each  station.  As  the  star  crosses  the  several 
threads  of  the  reticule  of  the  transit  instrument  at  the  eastern 
station,  the  observer,  by  means  of  a  break-circuit  key,  records 
the  instants  upon  both  chronographs.  The  same  process  is 
repeated  as  the  star  crosses  the  wires  at  the  western  station. 
Now,  it  is  evident  that  the  elapsed  time  between  the  transits  at 
the  two  meridians  has  been  recorded  upon  each  chronograph. 
Each  of  these  values  of  the  elapsed  time  is  to  be  corrected  for 
instrumental  errors,  errors  of  observation,  and  for  the  gain  or 
loss  of  the  clock  in  the  interval ;  and  the  mean  of  the  two 
values,  thus  corrected,  is  taken  as  the  difference  of  longitude  of 
the  two  places. 

By  making  similar  observations  on  several  stars  on  the  same 
night,  by  repeating  the  observations  on  subsequent  nights,  by 
exchanging  observers  and  using  different  clocks,  and,  finally, 
by  taking  a  mean  of  the  results,  a  very  accurate  determination 
of  the  difference  of  longitude  may  be  secured. 

87.  Method  of  Finding  the  Longitude  at  Sea. — The  method  of 
finding  the  longitude  at  sea  which  is  usually  employed  is  the 
method  of  Art.  82.  The  Greenwich  time  is  given  by  chro- 
nometers regulated  to  Greenwich  time,  and  the  local  time  is 
obtained  from  the  observed  altitudes  of  celestial  bodies.  The 
sun  is  the  body  the  altitude  of  which  is  most  commonly  used 
for  this  purpose;  but  altitudes  of  the  most  conspicuous  of  the 
placets  and  the  fixed  stars  may  also  be  successfully  employed. 
Altitudes  of  the  moon  are  to  be  avoided,  except  in  cases  where 
no  other  body  is  available.  At  the  instant  when  the  altitude 
of  any  celestial  body  is  observed,  the  time  shown  by  a  watch  is 
noted.  This  \vatch,  either  shortly  before  or  after  the  observa- 
tion, is  compared  with  the  Greenwich  chronometer,  and  by 
means  of  this  comparison  the  Greenwich  time  of  the  observa- 
tion is  obtained  from  the  time  given  by  the  watch.  The  neces- 
sary corrections  are  applied  to  the  sextant  reading  to  obtain 
the  body's  true  altitude.  We  shall  then  have,  in  the  triangle 
PZS,  Fig.  33,  the  side  ZS,  the  zenith  distance  of  the  body,  PS 
its  polar  distance,  obtained  from  the  Nautical  Almanac,  and 


COMPARISON   OF    LOCAL   TIMES.  81 

PZthe  co-latitude  of  the  place  of  observation ; 

the  latitude  being  determined  by  some  one  of 

the  methods  already  given  (Art.  79),  and  being 

reduced  to  the  time  of  observation  by  the  run  z>| 

of  the  ship  given  by  the  log.     In  the  triangle 

PZS,  then,  having  the  three  sides  given,  we 

can  compute  the  angle  SPZ,  which  is  the  hour 

angle  of  the  body.     From  this  hour  angle  the  Fig  33- 

local  time  can  be  readily  found,  from  which,  and  the  Greenwich 

time  already  obtained,  the  longitude  may  be  determined. 

In  case  there  is  no  chronometer  on  board,  the  method  of 
lunar  distances  is  the  only  regularly  available  method  of  deter- 
mining the  Greenwich  time.  At  the  present  day,  however, 
lunar  distances  are  mainly  employed  as  checks  upon  the  chro- 
nometer, since  any  change  in  the  rate  of  a  chronometer  will 
cause  a  discrepancy  between  the  Greenwich  time  shown  by  the 
chronometer  and  that  deduced  from  observation. 

It  can  be  shown,  by  proper  methods  of  investigation,  that 
an  error  in  the  assumed  latitude,  or  in  the  body's  altitude, 
causes  the  less  error  in  the  resulting  hour  angle  the  nearer 
the  body  is  to  the  prime  vertical.  It  is  best,  then,  in  observing 
the  altitude  of  any  celestial  body  for  the  purpose  of  obtaining 
the  local  time,  to  observe  it  wThen  the  body  bears  nearly  east  or 
west,  provided  the  altitude  is  not  so  small  as  to  be  sensibly 
affected  by  errors  in  the  refraction.  It  may  also  be  shown  that 
in  selecting  celestial  bodies  for  observations  of  this  character,  it 
is  best,  if  the  other  conditions  are  satisfied,  to  take  those  bodies 
which  have  the  smallest  declinations. 

88.  Comparison  of  the  Local  Times  of  Different  Meridians. — • 
Since  the  local  time,  either  solar  or  sidereal,  is  the  greater 
at  the  more  easterly  of  any  two  meridians,  it  follows  that  a 
watch  or  chronometer  which  is  regulated  to  the  time  of  any 
one  meridian  will  appear  to  gain  when  carried  to  the  west, 
and  to  lose  when  carried  to  the  east:  the  amount  of  gain  or 
loss  in  any  case  being  the  difference  of  longitude,  in  time, 
of  the  two  meridians.  A  watch,  for  instance,  which  gives  the 
correct  solar  time  at  Boston  will,  even  if  it  really  is  running 
accurately,  appear  to  gain  nearly  twelve  minutes  when  taken 


82  COMPARISON    OF    LOCAL   TIMES. 

to  New  York.  If,  then,  a  watch  which  is  regulated  to  the  solar 
time  of  any  meridian  is  carried  to  the  east,  the  difference  of 
longitude  in  time  between  the  meridian  left  and  that  arrived 
at  must  be  added  to  the  reading  of  the  watch,  to  obtain  tho 
time  at  the  second  meridian:  if  it  is  carried  to  the  west,  th<> 
difference  of  longitude  must  be  subtracted. 


ECLIPTIC. 


THE   SUN. 


CHAPTER   VI. 

THE  EARTH'S  ORBIT.      THE  SEASONS. 
THE  ZODIACAL  LIGHT. 


TWILIGHT. 


89.  The  Ecliptic. — IF  a  great  circle  on  any  globe  is  assumed 
to  represent  the  celestial  equator,  and  any  point  of  that  circle 
is  taken  to  represent  the  vernal  equinox,  the  relative  positions 
of  all  bodies,  the  right  ascension  and  declination  of  which  are 
known,  can  be  plotted  upon  this  globe,  and  we  shall  have  a 
representation  of  the  celestial  sphere.  The  poles  of  the  great 
circle  will  represent  the  poles  of  the  celestial  sphere,  and  all 
great  circles  passing  through  these  poles  will  represent  circles 
of  declination.  We  have  seen,  in  the  chapter  on  Astronomical 
Instruments,  in  what  manner  the  right  ascension  and  the  decli- 
nation of  any  celestial  body  can  be  determined  at  any  time  by 
observation.  If  we  thus  determine  the  position  of  the  sun  from 
day  to  day,  and  mark  the  corresponding  points  upon  our  celes- 
tial globe,  we  shall  find  that  the  sun  appears  to  move  in  a  great 
circle  of  the  sphere  from  west  to  east,  completing  one  revolution 
in  this  circle  in  365d.  6h.  9m.  9.6s.  of  our  ordinary  solar  time. 
This  interval  of  time  is  called  the  sidereal  year.  The  great  circle 
in  which  the  sun  appears  to  move  is  called  the  ecliptic,  and  the 
two  points  in  which  it  intersects  the  celestial  equator  are  called 
the  vernal  and  the  autumnal  equinox. 

Let  Fig.  34  be  a  representation  of 
the  celestial  sphere.  EA  Q  Fis  the  equi- 
noctial, Pp  is  the  axis  of  the  sphere,  and 
P  the  north  pole.  The  circle  A  CVD  re- 
presents  the  ecliptic,  V  the  vernal,  and 
A  the  autumnal  equinox.  The  sun  is 
at  the  vernal  equinox  on  the  21st  of 
March  It  thence  moves  eastward  and 
northward,  and  reaches  the  point  C, 


84-  DISTANCE   OF    THE   SFX. 

where  it  has  its  greatest  northern  declination,  on  the  2 1st  of  June. 
This  point  is  called  the  northern  summer  solstice.  From  this  point 
it  moves  eastward  and  southward,  passes  the  autumnal  equinox 
A  on  the  21st  of  September,  and  reaches  the  point  D,  called  the 
northern  winter  solstice,  on  the  21st  of  December.  It  thence 
moves  towards  V,  which  it  reaches  on  the  21st  of  t March. 

The  obliquity  of  the  ecliptic  to  the  equinoctial  is  the  angle 
CVty,  measured  by  the  arc  CQ.  This  angle  or  arc  is  evidently 
equal  to  the  greatest  declination,  either  north  or  south,  which 
the  sun  attains,  and  is  found  by  observation  to  be  about  23°  27'. 

90.  Definitions. — The  latitude  of  a  celestial  body  is  its  angular 
distance  from  the  plane  of  the  ecliptic,  measured  on  a  great  circle 
passing  through  its  poles,  and  called  a  circle  of  latitude.    In  Fig. 
34  the  arc  Ks  is  the  latitude  of  the  body  s.     The  longitude  of  a 
celestial  body  is  the  arc  of  the  ecliptic  intercepted  between  the 
vernal  equinox  and  the  circle  of  latitude  passing  through  the 
body.     Thus  VK  is  the  longitude  of  the  body  s.     Longitude  is 
properly  reckoned  towards  the  east. 

The  hour  circle  which  passes  through  the  solstices,  the  circle 
DHCB,  is  called  the  solstitial  colure.  The  hour  circle  which 
passes  through  the  equinoxes  is  called  the  equinoctial  colure. 

91.  Signs. — The  ecliptic  is  divided  into  twelve  equal  parts, 
called  signs t  which  begin  at  the  vernal  equinox,  and  are  named 
eastward  in  the  following  order:  Aries,  Taurus,  Gemini, Cancer, 
Leo,  Virgo,  Libra,  Scorpio,  Sagittarius,  Capricornus,  Aquarius, 
Pisces.     Hence  the  vernal  equinox  is  called  the  first  point  of 
Aries. 

The  Zodiac  is  a  zone  or  belt  on  the  celestial  sphere,  extend- 
ing about  9°  on  each  side  of  the  ecliptic. 

DISTANCE   OF    THE   SUN    FROM    THE    EARTH. 

92.  Relative  Distances  of  the  Earth  and  Venus  from  the  Sun. — 
It  is  found  by  observation  that  the  mean  value  of  the   sun's 
angular   semi-diameter   remains   constant   from    year    to   year, 
being  always  16'  2".     Since   any  increase   or  decrease   in   the 
distance  of  the  earth  from   the  sun  will  evidently  be   accom- 
panied by  a  corresponding  decrease   or  increase  in    the  sun's 
angular  semi-diameter,  we  conclude  that  the  mean  distance  of 


DISTANCE    OF    THE   SUN. 


85 


the  earth  from  the  sun  is  also  constant  from  year  to  year.  The 
distance  of  tne  earth  from  the  sun  is  obtained  by  determining 
the  sun's  horizontal  parallax  from  certain  observations  made 
upon  the  planet  Venus.  This  planet  revolves  in  a  nearly  cir- 
cular orbit  about  the  sun,  in  a  plane  only  3°  inclined  to  the 
plane  of  the  ecliptic.  Its  distance  from  the  sun  is  less  than 
that  of  the  earth  from  the  sun,  and  hence  it  sometimes  passes 
between  the  earth  and  the  sun,  and  is  seen  apparently  moving 
across  the  sun's  disc.  This  phenomenon  is  called  a  transit  of 
Venus.  As  a  preliminary  to  the  determination  of  the  earth's 
distance  from  the  sun  from  one  of  these  transits,  it  is  neces- 
sary to  obtain  the  relative  distances  of  Venus  and  the  earth 
from  the  sun.  To  do  this,  in  Fig.  35  let  S 
be  the  sun,  E  the  earth,  and  W  V"  V"  the 
orbit  of  Venus  about  the  sun.  It  is  evident 
that  the  greatest  angular  distance  (or  elon- 
gation) of  Venus  from  the  sun,  the  greatest 
value,  that  is,  of  the  angle  VES,  will  occur 
when  the  line  from  the  earth  to  Venus  is 
tangent  to  the  orbit  of  Venus,  as  repre- 
sented in  the  figure.  The  orbit  of  Venus  is 
not  really  a  circle,  but  an  ellipse,  and  hence 
the  distance  VS  is  slightly  variable.  So, 
also,  is  the  distance  SE;  hence  the  greatest 
elongation  is  also  variable,  being  found  to  lie  between  the  limits 
of  about  45°  and  47°.  Assuming  its  mean  value  to  be  46°,  we 
have  in  the  right-angled  triangle  VSE, 

VS  =  SE  sin  46°  —  .72  SE. 

Neglecting  the  inclination  of  the  orbit  of  Venus  to  the  plane 
of  the  ecliptic,  we  shall  have,  at  the  time  of  a  transit,  when 
Venus  is  at  V'» 

V'E  =  .28  SE. 

Hence  at  the  time  of  a  transit  the  distance  of  Venus  from  the 
sun  is  to  that  of  Venus  from  the  earth  as  about  72  to  28.* 

*  If  we  know  the  periodic  time  of  Venus  and  that  of  the  earth,  the  ratio 
of  the  distances  of  these  two  planets  from  the  sun  can  be  obtained  by  Kepler's 
Third  Law  (Art.  117),  that  "the  squares  of  the  periodic  times  of  any  two 
planet*  are  proportional  to  the  cubes  of  their  mean  distances  from  the  sun." 

8 


86  DISTANCE    OF    THE   SUN. 

93.   Transit  of  Venus. — In  Fig. .36,  let  S  denote  the  centre  of 
the  sun,  and  CADN  its  disc :  let  V  be  Venus,  and  E  the  centre 


Fig.  36. 

of  the  earth.  Let  HK  be  that  diameter  of  the  earth  which  is 
perpendicular  to  the  plane  of  the  ecliptic,  and  let  an  observer  be 
supposed  to  be  stationed  at  each  extremity.  In  order  to  simplify 
the  explanation,  let  us  neglect  the  rotation  of  the  earth  during 
the  observation,  and  suppose  Venus  to  move  in  the  plane  of  the 
ecliptic.  To  the  observer  at  If,  Venus  will  appear  to  move 
across  the  sun's  disc  in  the  chord  CD,  and  to  the  observer  at  JT, 
in  the  chord  AB.  Regarding  VHK  and  VFG  as  similar  tri- 
angles, we  have,  by  Geometry, 

FG:HK=  GV:  7ff  =  72:28 


=-  HK. 

Again,  we  can  obtain  the  angle  which  the  line  FG  subtends 
at  the  earth's  centre  in  the  following  manner.  Let  the  observer 
at  H  note  the  interval  of  time  in  which  the  planet  crosses  thf» 
sun's  disc  in  the  chord  CD,  and  the  observer  at  K  HIQ  interval 
in  which  it  moves  through  the  line  AB.  Since  there  are  tables 
which  give  us  the  angular  velocity,  as  seen  from  the  earth,  both 
of  the  sun  and  of  Venus,  we  can  deduce  the  angles  at  the  earth's 
centre  subtended  by  the  chords  FB  and  GD,  and  knowing  also 
the  angular  semi-diameter  of  the  sun,  in  other  words,  the  angle 
at  the  earth's  centre  subtended  by  SB  or  SD,  we  can  compute 
the  angles  at  the  earth's  centre  subtended  by  FS  and  GSt  and, 
finally,  the  angle  subtended  by  FG. 

We  have  now  determined  the  angle  subtended  by  the  line  FG, 
at  a  distance  equal  to  that  of  the  earth  from  the  sun,  and  also 
the  ratio  of  FG  to  the  earth's  diameter.  It  is  evidently  easy  to 
obtain  from  these  values  the  angle  at  the  sun  subtended  by  the 


.MAGNITUDE    OF    THE   SUN.  87 

earth's  radius,  which  angle  is  the  sun's  horizontal  parallax,  ;u* 
we  have  already  seen  in  Art.  54. 

Although  we  have  assumed  in  this  discussion  that  the  two  ob- 
servers are  stationed  at  the  extremities  of  the  same  diameter,  it 
is  really  only  necessary  that  they  shall  be  at  two  places  whose 
difference  of  latitude  is  large.  The  earth's  rotation  and  other 
things  which  we  have  here  neglected  must  be  taken  into  con- 
sideration in  the  practical  determination  of  the  sun's  parallax. 

94.  Distance  of  the  Earth  from  the  Sun. —  The  last  two  transits 
of  Venus  were  in  1769  and  1874,  and  from  observations  in  1769, 
the  sun's  horizontal  parallax  was  determined  to  be  8". 6.     Later 
observations  of  a  different  character  have  given  a  horizontal 
parallax  of  8".848,*  which  is  here  used.     The  results  of  the  exten- 
sive observations  in  1874  cannot  yet  be  given. 

From  Art.  54,  we  have  for  the  distance  in  miles  of  the  earth 
from  the  sun, 

d  =  E  cosec  P  =  3962.8  cosec  8".848  =  92,400,000  miles. 

95.  Magnitude  of  the  Sun. — 
The  length  of  the  sun's  radius 
can  be  at  once  obtained  as  soon 
as  we  know  its  distance  from 
the  earth.     Thus,  in  Fig.  37, 

let  S  be  the  centre  of  the  sun,  ris-  37- 

and  E  that  of  the  earth.  The  angle  AES  is  the  apparent  semi- 
diameter  of  the  sun,  which  we  obtain  by  observation,  its  mean 
value  being,  as  already  stated,  16'  2".  We  have  then,  in  the 
right-angled  triangle  AES, 

SA  =  92,400,000  sin  16'  2"  =  431,000  miles. 
The  sun's  linear  radius,  then,  is  equal  to  nearly  109  of  the  earth's 
radii;  and  since  the  volumes  of  spheres  are  proportional  to  the 
cubes  of  their  radii,  the  volume  of  the  sun  bears  to  that  of  the 
earth  the  enormous  ratio  of  1,286,000  to  1. 

By  observations  and  calculations  which  will  be  described  in 
fhe  Chapter  on  Gravitation  (see  Art.  114),  the  mass  of  the  sun  is 
found  to  be  about  327,000  times  that  of  the  earth;  or  about  670 
times  the  sum  of  ihe  masses  of  all  the  planets  of  the  solar  system. 

*  Determination  of  Professor  Simon  Newcomb,  Tjnited  States  Navy.  TU 
value  obtained  by  Leverrier  is  8x/.95. 


88 


ORBIT   OF    THE    EARTH. 


THE  EARTH'S  ORBIT. 

96.  Revolution  of  the  Earth  about  the  Sun. — Up  to  this  point 
we  have  spoken  of  the  apparent  animal  motion  of  the  sun  in  the 
ecliptic  from  west  to  east,  as  though  the  earth  were  really  at  rest, 
and  the  sun  revolved  about  it  in  its  orbit.  But  when  we  take 
into  consideration  the  immense  mass  of  the  sun  compared  with 
that  of  the  earth,  we  are  almost  irresistibly  led  to  conclude  that 
the  apparent  annual  revolution  of  the  sun  is  the  result,  not  of  the 
actual  revolution  of  the  sun  about  the  earth,  but  of  that  of  the 
earth  about  the  sun.  Such  a  revolution  of  the  earth,  from  west 
to  east,*  would  give  to  the  sun  precisely  that  apparent  motion 

in  the  ecliptic  which  has  been  ob- 
served. This  may  be  seen  in  Fig. 
38.  Let  S  be  the  sun,  EE'E"  the 
earth's  orbit,  and  the  outer  circle 
S'S"S'"  the  great  circle  in  which 
the  plane  of  the  ecliptic,  indefi- 
nitely extended,  meets  the  celestial 
sphere.  When  the  earth  is  at  E, 
the  sun  will  be  projected  in  $'; 
when  the  earth  is  at  E',  the  sun 
will  be  projected  in  S",  &c. ;  that 
is  to  say,  while  the  earth  moves 


Fig.  38. 


*  Whatever  the  absolute  motion  of  any  celestial  body  moving  in  a  circle 
or  an  ellipse  may  be,  the  appearance  presented  in  that  motion  will  be  re- 
versed if  the  spectator  moves  from  one  side  of  the  plane  in  which  the 
motion  is  performed  to  the  other.  Thus,  the  apparent  daily  motion  to  the 
westward  of  any  celestial  body  is  the  same  as  the  motion  of  the  hands  of  a 
clock  as  we  look  upon  its  face,  to  an  observer  who  is  on  the  north  side  of 
the  plane  of  the  diurnal  circle  in  which  the  body  moves,  as  is  seen  in  any 
latitude  north  of  23°  27'  in  the  motion  of  the  sun  ;  while  the  same  west- 
ward motion  presents  the  opposite  appearance  if  the  ohserveristo  the  south 
of  the  plane  of  motion,  as  may  be  seen  in  these  latitudes  in  the  case  of  the 
Great  Bear.  The  appearance  presented  by  a  motion  from  west  to  east  is 
of  course  the  reverse  of  this ;  hence  when  we  say  that  the  earth  or  any 
other  body  moves  about  the  sun  from  west  to  east,  we  mean  that,  to  an  ob- 
server situated  to  the  north  of  the  plane  of  motion,  the  body  appears  to  movein  a 
direction  opposite  to  that  in  which  the  hands  of  a  dock  move. 


ORBIT   OF   THE   EARTH.  89 

about  the  sun  in  the  direction  EE'E",  the  sun  will  apparently 
move  about  the  earth  in  the  same  direction,  S'S"S'". 

Against  this  theory,  then,  of  the  earth's  revolution  there  is 
nothing  to  urge;  and  analogy  gives  us  a  strong  argument  in 
favor  of  it.  Almost  every  celestial  body  in  which  any  motion 
at  all  can  be  detected  is  found  to  be  revolving  about  some  other 
body,  larger  than  itself.  The  moon  revolves  about  the  earth; 
the  satellites  of  the  planets  revolve  about  the  planets  ;  and  the 
planets  themselves,  some  of  which  are  much  larger  than  the 
earth,  and  at  a  much  greater  distance  from  the  sun,  revolve 
about  the  sun.  Henceforward,  then,  we  shall  include  the  earth 
in  the  list  of  planets,  and  consider  the  sidereal  year  to  be  -the 
interval  of  time  in  which  the  earth  makes  one  complete  revolu- 
tion about  the  sun. 

97.  Linear  Velocity  of  the  Earth  in  its  Orbit.  —  The  number  of 
miles  in  the  circumference  of  the  earth's  orbit,  considered  as  a 
circle,  is  obtained  by  multiplying  the  radius  of  the  orbit  by  2*. 
If  we  then  divide  this  product  by  the  number  of  seconds  in  a 
year,  we  shall  have,  in  the  quotient,  the  number  of  miles  through 
which  the  earth  moves  about  the  sun  in  a  second  of  time.     It 
will  be  found  to  be  about  18.4  miles. 

98.  Elliptical  Form  of  the  Earth's   Orbit.—  Although,  as  has 
already  been  stated,  the  mean  value  of  the  sun's  angular  semi- 
diameter  remains  constant  from  year  to  year,  careful  measure- 
ments of  the  semi-diameter  show  that  it  varies  in  magnitude 
during  the  year,  being  greatest  about  the  first  of  January,  and 
least  about  the  first  of  July.     The  evident  conclusion  from  this 
fact  is  that  the  distance  between  the  earth  and  the  sun  also  varies 
during  the  year,  being  greatest  when  the  sun's  semi-diameter  is 
the  least,  and  least  when  it  is  the  greatest.     The  truth  of  this 
conclusion  may  be  seen  in  Fig.  37,  in  which  we  have 

AS 

sm 


As  AS  of  course  remains  constant,  ES  will  vary  inversely  as  sin 
AE/S,  or  since  the  sines  of  small  angles  are  proportional  to 
the  angles  themselves,  inversely  as  the  angle  AES  itself.  The 
angular  semi-diameter  of  the  sun  is  16'  17".  8,  the  least 


90  ORBIT   OF   THE   EARTH. 

is  Jo'  45".5:  hence  the  ratio  of  tfce  greatest  to  the  least  distance 
is  that  of  16'  17".8  to  15  45".5,  or  of  1.034  to  1. 

Let  us  now  assume  any  line,  SA  in  Fig.  39,  for  instance,  as  our 
unit  of  measure,  and  prolong  it  until 
SH  is  to  SA  as  1.034  is  to  1.  Then  if  S 
denotes  the  sun,  SA  and  SH  will  repre- 
sent the  relative  distances  of  the  earth 
from  the  sun  on  about  the  first  of  Janu- 
ary and  the  first  of  July.  On  certain 

•^ ^D  days  throughout  the  year,  let  the  advance 

Fir  39.  of  the  sun  in  longitude  since  the  time 

when  the  earth  was  at  A  be  determined,  and  let  the  angular 
semi-diameter  of  the  sun  on  each  of  these  days  be  measured. 
Lay  off  the  angles  ASB,  ASC,  &c.,  equal  to  these  advances  in 
longitude.  Since,  as  may  readily  be  seen  in  Fig.  38,  the  appa- 
rent advance  of  the  sun  in  longitude  is  caused  by  the  advance 
of  the  earth  in  its  orbit,  and  is  equal  to  it,  the  angles  ASB,  ASCt 
&c.,  will  represent  the  angular  distances  of  the  earth  from  the 
point  A  on  the  days  when  the  different  observations  were  made. 
Let  us  next  take  the  lines  SB,  SC,  &c.,  of  such  lengths  that 
each  line  may  be  to  SA  in  the  inverse  ratio  of  the  corresponding 
semi-diameters.  If,  then,  we  draw  a  line  through  the  points 
A,  B,  C,  &c.,  we  shall  have  a  representation  of  the  orbit  of  the 
earth  about  the  sun.  The  curve  is  found  to  be  an  ellipse,  the 
sun  being  at  one  of  the  foci.  The  point  A,  where  the  earth  is 
nearest  to  the  sun,  is  called  the  perihelion,  the  point  H,  the  aphe- 
lion; and  the  angular  distance  of  the  earth  from  its  perihelion 
is  called  its  anomaly. 

The  eccentricity  of  the  ellipse,  or  if  0  is  the  centre  of  the 
ellipse,  the  ratio  of  OS  to  OA,  is  evidently  equal  to  about  T?oV?> 
or  -g-^th.  A  more  accurate  value  of  it  is  .0167917.  This  eccen- 
tricity is  at  present  subject  to  a  diminution  of  .000041  a 
century ;  but  Leverrier,  a  French  astronomer,  has  proved  that 
after  the  eccentricity  has  diminished  to  a  certain  point  it  w'H 
begin  to  increase  again. 

THE  SEASONS. 

99.  The  change    uf  seasons  on  the  earth  is  caused  by  the 


SEASONS. 


91 


inequality  of  the  days  and  nights,  and  this  inequality  is  a  result 
of  the  inclination  of  the  plane  of  the  equinoctial  to  that  of  the 
ecliptic.  The  relative  positions  of  the  sun  and  the  earth  at  dif- 
ferent parts  of  the  year  are  represented  in  Fig.  40.  S  represents 


Fig.  40. 

the  sun,  and  ABCD  the  orbit  of  the  earth.  Pp  is  the  axis  of 
rotation  of  the  earth,  and  EQ  the  equator.  The  plane  of  this 
equator  is  supposed  to  intersect  the  plane  of  the  ecliptic  in  the 
line  of  equinoxes  A  C.  Since,  as  we  have  already  seen,  the  sun 
appears  to  be  on  this  line  on  the  21st  of  March  and  the  21st  of 
September,  the  earth  itself  must  also  be  on  this  line  at  the  same 
time.  Suppose,  then,  the  earth  to  be  at  A  on  the  21st  of  March. 
The  sun  will  evidently  lie  in  the  direction  AS,  and  will  be  pro- 
jected on  the  celestial  sphere  at  the  vernal  equinox.  Now,  since 
a  line  which  is  perpendicular  to  a  plane  is  perpendicular  to  every 
line  in  that  plane  which  is  drawn  to  meet  it,  the  axis  Pp  at  A, 
being  perpendicular  to  the  equator,  is  also  perpendicular  to  the 
line  AS,  which  is  common  to  both  the  plane  of  the  equator  and 


92  SEASONS. 

the  plane  of  the  ecliptic.  Half  of  each  parallel  of  latitude  on 
the  earth  will  therefore  lie  in  light  and  half  in  darkness ;  and 
hence,  as  the  earth  rotates  on  the  axis  Pp,  every  point  on  its 
surface  will  describe  half  of  its  diurnal  course  in  light  and 
half  in  darkness :  in  other  words,  day  and  night  will  be  equal 
over  the  whole  earth.  Since  the  direction  of  the  axis  of  rota- 
tion remains  unchanged,  the  same  condition  of  things  will  occur 
when  the  earth  is  at  C,  on  the  21st  of  September.  Let  the 
earth  be  at  B  on  the  21st  of  June.  Here  we  see  that,  as  the 
earth  rotates  on  its  axis  Pp,  every  point  on  its  surface  within 
the  circle  ab  will  lie  continually  in  the  light,  and  will  hence 
have  continual  day,  while  within  the  corresponding  circle  a'b' 
the  night  will  be  continual.  We  see  also  that  at  the  equator 
the  days  and  nights  will  be  equal,  and  that  every  point  between 
the  equator  and  the  circle  ab  will  describe  more  of  its  diurnal 
course  in  light  than  in  darkness,  and  will  thus  have  its  days 
longer  than  its  nights ;  while  between  the  equator  and  the  circle 
ab'  the  nights  will  be  longer  than  the  days.  Similar  phenomena 
will  occur  when  the  earth  is  at  D,  on  the  21st  of  December, 
except  only  that  it  will  then  be  the  southern  hemisphere  in 
which  the  days  .are  longer  than  the  nights,  and  the  southern 
pole  at  which  the  sun  is  continually  visible. 

Such,  then,  is  the  inequality  of  the  days  and  nights  caused 
by  the  inclination  of  the  plane  of  the  equinoctial  to  that  of  the 
ecliptic.  As  the  sun  apparently  moves  from  either  equinox, 
the  inequality  of  day  and  night  continually  increases,  reaches 
its  maximum  when  the  sun  arrives  at  either  solstice,  and  then 
continually  decreases  as  the  sun  moves  on  to  the  equinox :  the 
day  being  longer  than  the  night  in  that  hemisphere  which  is 
on  the  same  side  of  the  equator  with  the  sun.  Now,  any  point 
on  the  earth's  surface  receives  heat  during  the  day  and  radiates 
it  during  the  night :  and  hence,  when  the  days  are  longer  than 
the  nights,  the  amount  of  heat  received  is  greater  than  the 
amount  radiated,  and  the  temperature  increases ;  while,  on  the 
contrary,  when  the  days  are  shorter  than  the  nights,  the  tempe- 
rature decreases :  and  thus  is  brought  about  the  change  of  sea- 
eons  on  the  earth. 

Another  fact,  depending  on  the  same  cause,  and  tending  to 


SEASONS.  93 

the  same  result,  must  also  be  taken  into  consideration;  and 
that  is  that  the  temperature  at  any  place  depends  on  the  ob- 
liquity of  the  sun's  rays :  on  the  altitude,  in  other  words,  which 
the  sun  attains  at  noon.  Now  we  have,  from  Art.  76, 

z  =  L  —  d: 

from  which  we  see  that,  the  latitude  remaining  constant,  the  sun 
attains  the  greater  altitude,  the  greater  its  declination  when  it 
has  the  same  name  as  the  latitude,  and  the  less  its  declination 
when  it  has  the  opposite  name :  so  that  the  nearest  approach  to 
vertically  in  the  sun's  rays  will  occur  at  the  same  time  that 
the  day  is  the  longest.  An  exception,  however,  must  be  noticed 
to  this  general  rule,  in  the  case  of  places  within  the  tropics : 
since  at  these  places,  as  may  be  seen  from  the  formula,  the  sun 
passes  through  the  zenith  when  its  declination  is  equal  to  the 
latitude,  and  has  the  same  name. 

100.  Effect  of  the  Ellipticity  of  the  Earths  Orbit  on  the  Change 
of  Seasons. — The  elliptic  form  of  the  earth's  orbit  has  very 
little  to  do  with  the  change  of  seasons.  For  although  the  earth 
is  nearer  to  the  sun  on  the  1st  of  January  than  on  the  1st  of 
July,  yet  its  angular  velocity  at  that  time  is  found  by  observa- 
tion to  be  greater,  and  to  vary  throughout  the  whole  orbit  in- 
versely as  the  square  of  the  distance.  Now  it  may  readily  be 
shown  that  the  amount  of  heat  received  by  the  earth  at  different 
parts  of  its  orbit  also  varies,  other  things  being  equal,  inversely 
as  the  square  of  the  distance :  so  that  equal  amounts  of  heat 
are  received  by  the  earth  in  passing  through  equal  angles  of 
its  orbit,  in  whatever  part  of  its  orbit  those  angles  may  be 
situated.  Still,  although  the  change  in  distance  does  not  mate- 
rially affect  the  annual  change  of  seasons,  it  does  affect  the 
relative  intensities  of  the  northern  and  the  southern  summer. 
The  southern  summer  takes  place  when  the  earth's  distance  is 
only  about  f|jths  of  what  it  is  at  the  time  of  the  northern 
summer:  hence,  the  intensity  at  the  former  period  will  be  to 
that  at  the  latter  in  the  ratio  of  about  (|$)2  to  1,  or  about  }|  to 
1 :  in  other  words,  the  intensity  of  the  heat  of  the  southern  sum- 
mer will  be  y^th  greater  than  that  of  the  heat  of  the  northern 
summer. 


94  TWILIGHT. 


TWILIGHT. 

101.  If  the  earth's  atmosphere  did  not  contain  particles  of 
dust  and  vapor,  which  serve  to  reflect  the  rays  of  light,  the 
transition  from  day  to  night  would  be  instantaneous,  and  the 
intermediate  phenomenon  of  twilight  would  have  no  existence. 

This  phenomenon 
is  explained  in 
Fig.  41,  in  which 
ABC  represents 

B<^         /  x\  a  portion  of  the 

earth's      surface, 

Fig-  4i.  &    and  EDF  a  por- 

tion of  the  atmosphere.  Let  the  sun  be  supposed  to  lie  in  the 
direction  AS,  and  to  be  in  the  horizon  of  the  place  A.  All  of  the 
atmosphere  which  lies  above  the  horizontal  plane  SD  will  then 
receive  the  direct  rays  of  the  sun,  and  A  will  receive  twilight 
from  the  whole  sky.  The  point  B  will,  on  the  contrary,  be  illu- 
minated only  by  the  smaller  portion  of  the  atmosphere  included 
within  the  planes  EB  and  AD  and  the  curved  surface  ED;  and 
at  the  point  C  the  twilight  will  have  wholly  ceased.  Strictly 
speaking,  the  lines  AS,  BE,  &c.,  should  be  slightly  curved,  owing 
to  the  effects  of  refraction,  but  the  omission  involves  no  change 
in  the  explanation. 

It  is  computed  that  twilight  ceases  when  the  sun  is  about  18° 
below  the  horizon,  measured  on  a  vertical  circle.  The  more 
nearly  perpendicular  to  the  horizon  is  the  diurnal  circle  in 
which  the  sun  appears  to  move,  the  more  rapid  will  be  the  sun's 
descent  below  the  horizon ;  hence,  the  length  of  twilight  dimin- 
ishes as  we  approach  the  equator  and  increases  as  we  recede 
from  it.  Furthermore,  we  see  in  Fig.  2  that  the  greater  the 
declination  of  the  sun,  the  smaller  is  the  apparent  diurnal  circle 
in  which  it  moves,  and  the  greater  will  be  the  length  of  time 
required  for  the  sun  to  reach  the  depression  of  18°  below  the 
horizon.  The  shortest  twilight,  therefore,  occurs  at  places  on  the 
equator,  when  the  sun  is  on  the  equinoctial,  and  its  length  is 
then  Ih.  12m.  Near  the  poles  the  length  of  twilight  is  at  times 
very  great.  Dr.  Hayes,  in  his  last  expedition  towards  the  North 


APPEARANCE    OF   THE   SUN.  95 

P^le,  wintered  at  latitude  78°  18'  N.,  so  far  above  the  circle  ab, 
Fig.  40,  that  the  sun  was  continually  below  the  horizon  from  the 
noddle  of  October  to  the  middle  of  February;  but  at  the  begin- 
ning and  the  end  of  this  interval  twilight  lasted  for  about  nine 
hours.  At  the  poles  twilight  lasts  nearly  a  month  and  a  half. 

GENERAL   DESCRIPTION    OF    THE   SUN. 

102.  When  the  sun  is  observed  with  a  telescope,  spots  are 
noticed  upon  its  surface.  These  spots  appear  to  cross  the  sun's 
disc  from  east  to  west,  and  with  different  rates,  the  rate  of 
motion  of  spots  at  the  sun's  equator  being  the  greatest.  We 
therefore  conclude  that  these  spots  have  not  only  an  apparent 
rsotion,  caused  by  the  sun's  rotation,  but  also  a  proper  motion 
of  their  own.  By  appropriate  investigation  of  these  motions, 
it  is  found  that  the  sun  rotates  from  west  to  east  upon  a  fixed 
axis,  in  a  plane  inclined  at  an  angle  of  about  7°  to  the  plane  of 
the  ecliptic.  The  period  of  this  rotation  is  about  25  days. 

Much  uncertainty  exists  as  to  the  nature  of  these  spots.  Un- 
til recently,  it  was  held  that  the  sun  is  surrounded  by  two 
atmospheres,  of  which  only  the  outer  one  (called  the  photo- 
sphere} is  luminous,  and  that  the  spots  are  rents  in  these 
atmospheres  through  which  the  solid  body  of  the  sun  is  seen. 
These  spots  are  for  the  most  part  confined  to  a  zone,  extending 
about  35°  on  each  side  of  the  sun's  equator.  They  differ  widely 
iii  duration,  sometimes  lasting  for  several  months,  and  some- 
times disappearing  in  the  course  of  a  fe^  hours.  They  are 
sometimes  of  an  immense  size.  One  was  seen  in  1843,  with 
a  diameter  of  nearly  75,000  miles :  it  remained  in  sight  for 
a  week,  and  was  visible  to  the  naked  eye.  In  1858,  a  much 
larger  one  was  seen,  its  diameter  being  over  140,000  miles.  As 
a  general  thing,  each  dark  spot,  or  umbra,  as  it  is  called,  has 
within  it  a  still  darker  point,  called  the  nucleus,  and  is  sur- 
rounded by  a  fringe  of  a  lighter  shade,  called  the  penumbra. 
Sometimes  several  spots  are  inclosed  by  the  same  penumbra; 
and  occasionally  spots  are  seen  without  any  penumbra  at  all. 

On  the  theory  of  two  atmospheres,  the  existence  of  the 
penumbra  is  explained  by  supposing  the  aperture  in  the  outer 
and  luminous  stratum  to  be  wider  fb;.n  that  in  the  inner  one, 


96  APPEARANCE    OF    THE    SUN. 

and  that  portions  of  the  inner  ^tratum,  being  subjected  to  a 
strong  light  from  above,  are  rendered  visible:  the  umbra  itself 
being,  as  already  remarked,  the  solid  body  of  the  sun  seen  through 
both  strata.  According  to  Mr.  J.  N.  Lockyer,  "sun-spots  are 
cavities  qr  hollows  eaten  into  the  photosphere,  and  these  different 
shades  [the  penumbra,  umbra,  and  nucleus]  represent  different 
depths."  [See  Note,  page  99.] 

One  very  curious  and  interesting  discovery  in  relation  to  these 
spots  is  that  of  a  periodicity  in  their  number.  This  discovery 
was  made  by  Schwabe,  of^Dessau,  whose  researches  and  obser- 
vations on  this  subject  covered  a  period  of  more  than  twenty-five 
years.  The  number  of  groups  of  spots  which  he  observed  in  a 
year  varied  from  33  to  333,  the  average  being  not  far  from  150.* 
He  found  the  period  from  one  maximum  to  another  to  be  about 
ten  years.  Professor  Wolf,  of  Zurich,  after  tabulating  all  the  ob- 
servations of  spots  since  1611,  decided  that  the  period  varied  from 
eight  to  sixteen  years,  its  mean  value  being  about  eleven  years. 
Recent  investigations  show  that  this  periodicity  is  in  some  way 
connected  with  the  action  of  the  planets,  of  Jupiter  and  Venus 
particularly,  upon  the  sun's  photosphere.  It  is  a  curious  fact  that 
magnetic  storms  and  the  phenomenon  called  Aurora  or  Northern 
Lights  have  a  similar  period,  and  are  most  frequent  and  most 
striking  when  the  number  of  the  solar  spots  is  the  greatest. 

Still  other  phenomena  which  are  seen  upon  the  sun's  disc  are 
the  faculce,  which  are  streaks  of  light  seen  for  the  most  part  in 
the  region  of  the  spots,  and  which  are  undoubtedly  elevations  or 
ridges  in  the  photosphere:  and  the  luculi,  which  are  specks  of 
light  scattered  over  the  sun's  disc,  giving  it  an  appearance  not 
unlike  that  of  the  skin  of  an  orange,  though  relatively  much  less 
rough.  The  cause  of  these  lucuii  is  unknown. 

At  the  time  of  a  total  eclipse  of  the  sun  by  the  moon,  the  disc 
of  the  sun  is  observed  to  be  surrounded  by  a  ring  or  halo  of  light, 
which  is  called  the  corona.  The  breadth  of  this  corona  is  more 
than  equal  to  the  diameter  of  the  sun.  Many  theories  have  been 
advanced  to  explain  this  phenomenon,  one  of  which  is  that  it  is 
due  to  the  existence  of  still  another  atmosphere,  exterior  to  the 

*  A  table  of  Schwnbe's  observations  is  given  in  the  Appendix. 


CONSTITUTION    OF    THE   SUN.  97 

photosphere.  Another  theory  is  that  this  corona  consists  of 
streams  of  luminous  matter,  radiating  in  all  directions  from  the 
sun.  Rose-colored  protuberances,  sometimes  called  red  flamea, 
are  also  seen,  which  are  usually  of  a  conical  shape,  and  are 
sometimes  of  great  height.  In  the  total  eclipse  of  August  17th, 
1868,  one  was  observed  with  an  apparent  altitude  of  3',  corre- 
sponding to  a  height  of  about  80,000  miles.  These  protuberances 
were  formerly  supposed  to  be  similar  in  character  to  our  terres- 
trial clouds;  but  Dr.  Jannsen,  the  chief  of  the  French  expedi- 
tion sent  out  to  the  East  to  observe  the  total  eclipse  of  August, 
1868,  examined  their  light  with  the  spectroscope,  and  found  them 
to  be  masses  of  incandescent  gas,  of  which  the  greater  part  was 
hydrogen.  Dr.  Jannsen  also  made  the  interesting  discovery  that 
these  protuberances  can  be  examined  at  any  time,  without  wait- 
ing for  the  rare  opportunity  afforded  by  a  total  eclipse.  He 
observed  them  for  several  successive  days,  and  found  that  great 
changes  took  place  in  their  form  and  size.  Mr.  Lockyer,  of 
England,  who  has  since  examined  them,  pronounces  them  to  be 
merely  local  accumulations  of  a  gaseous  envelope  completely 
surrounding  the  sun :  the  spectrum  peculiar  to  these  protuber- 
ances appearing  at  all  parts  of  the  disc. 

It  has  already  been  stated  (Art.  47)  that  the  spectroscope  en- 
ables us  to  establish  the  existence  of  certain  chemical  substances 
in  the  sun,  by  a  comparison  of  the  spectra  of  these  substances 
with  that  of  the  sun ;  or,  more  precisely,  by  a  comparison  of  the 
lines,  bright  or  dark,  by  which  these  different  spectra  are  dis- 
tinguished. The  number  of  the  parallel  dark  lines  in  the  solar 
spectrum  which  have  been  detected  and  mapped  exceeds  3000; 
and  careful  examination  also  shows  that  some  of  these  are  double. 
3ome  of  the  more  prominent  of  these  lines  have  received  the 
names  of  the  first  letters  of  the  alphabet;  D,  for  example,  is  a 
very  noticeable  double  line  in  the  orange  of  the  spectrum.  When 
certain  chemical  substances  are  evaporated,  either  in  a  flame  or 
by  the  electric  current,  the  spectra  which  they  form  are  also 
characterized  by  lines,  which,  however,  are  not  dark,  but  bright. 
If,  for  instance,  sodium  is  introduced  into  a  flame,  its  incan- 
descent vapor  produces  a  spectrum  which  is  characterized  by  a 
brilliant  double  band  of  yellow;  and  it  is  especially  noticeable 


98  CONSTITUTION    OF    THE   SUN. 

that  this  yellow  band  coincides,  exactly  in  position  with  the 
dark  line  D  of  the  solar  spectrum.  In  the  same  way  the 
spectrum  of  zinc  is  found  to  contain  bands  of  red  and  blue- 
that  of  copper  contains  bands  of  green:  and,  in  general,  the 
spectrum  of  each  metal  contains  certain  bright  bands  or  lines, 
peculiar  to  itself,  and  readily  recognized.  We  may  therefore 
conclude  that  an  incandescent  gas  or  vapor  emits  rays  of  a  certain 
refrangibility  and  color,  and  those  rays  only. 

Again,  it  is  proved  by  experiment  that  if  a  ray  of  white  light 
be  allowed  to  pass  through  an  incandescent  vapor,  the  vapor 
will  absorb  precisely  those  rays  which  it  can  itself  emit.  If,  for 
instance,  a  continuous  spectrum  be  formed  by  a  ray  of  intense 
white  light  from  any  source,  and  if  the  vapor  of  sodium  be  intro- 
duced in  the  path  of  this  ray,  between  the  prism  and  the  source 
of  light,  a  dark  band  will  appear  in  the  spectrum,  identical  in 
position  with  the  bright  yellow  band  which  we  have  already 
noticed  in  the  spectrum  of  sodium,  and  which  we  found  to  be 
identical  in  position  with  the  dark  line  D  of  the  solar  spectrum. 

We  are  now  ready  to  apply  the  principles  established  by 
these  experiments  to  the  case  of  the  sun.  The  sun  is,  as  we 
saw  above,  a  sphere  surrounded  by  a  vaporous  envelope.  This 
sphere  would  of  itself  emit  all  kinds  of  rays,  and  therefore 
give  a  continuous  spectrum  ;  but  the  photosphere  which  sur- 
rounds it  absorbs  those  of  the  sun's  rays  which  it  can  itself 
emit.  The  dark  line  D  of  the  solar  spectrum  shows,  as  in  the 
experiment  above  described,  that  sodium  has  been  introduced 
in  the  path  of  the  sun's  rays :  in  other  words,  that  sodium  is  in 
the  sun's  photosphere.  In  the  same  way,  Professor  Kirchhoff, 
to  whom  we  owe  this  remarkable  discovery,  has  established  the 
existence  in  the  photosphere  of  iron,  calcium,  magnesium,  chro- 
mium, and  other  metals.  In  the  case  of  iron,  more  than  450 
bright  lines  have  been  detected  in  its  spectrum :  and  for  every 
one  of  these  lines  there  is  a  corresponding  dark  line  in  the  solar 
spectrum. 

We  also  see,  from  the  preceding  experiments,  how  the  presence 
of  bright  lines  in  the  spectrum  of  the  rose-colored  protuberances 
could  prove  to  Dr.  Jannsen  that  these  protuberances  were  not 
masses  of  cloud.?,  reflecting  the  light  of  the  sun,  but  masses  of 


ZODIACAL    LIGHT.  99 

ircandescent  vapor.     We  shall  see  another  instance  of  the  same 
description  when  we  come  to  examine  some  of  the  nebulae. 

THE    ZODIACAL    LIGHT. 

103.  At  certain  seasons  of  the  year  a  faint  nebulous  light, 
not  unlike  the  tail  of  a  comet,  is  seen  in  the  west  after  twilight 
has  ended,  or  in  the  east  before  it  has  begun.  This  is  called 
the  Zodiacal  Light.  Its  general  shape  is  nearly  that  of  a  cone, 
the  base  of  which  is  turned  towards  the  sun.  The  breadth  of 
the  base  varies  from  8°  to  30°  of  angular  magnitude.  The  apex 
of  the  cone  is  sometimes  more  than  90°  to  the  rear  or  in  advance 
of  the  sun.  According  to  Humboldt,  it  is  almost  always  visible, 
at  the  times  above  stated,  within  the  tropics:  in  our  latitudes  it 
is  seen  to  the  best  advantage  in  the  evening  near  the  first  of 
March,  and  in  the  morning  near  the  middle  of  October. 

Of  the  many  theories  proposed  to  account  for  the  zodiacal 
light,  the  one  which  seems  to  be  most  widely  accepted  is  that  it 
consists  of  a  ring  or  zone  of  rare  nebulous  matter  encircling 
the  sun,  which  reaches  as  far  as  the  earth,  and  perhaps  extends 
beyond  it.  According  to  another  theory,  it  is  a  belt  of  meteoric 
bodies  surrounding  the  sun.  A  very  interesting  and  valuable 
series  of  observations  upon  the.  Zodiacal  Light  was  made  by 
Chaplain  Jones,  United  States  Navy,  in  the  years  1853-5,  in 
latitudes  ranging  from  41°  N.  to  53°  S,  The  conclusion  which 
he  drew  from  his  observations  was  that  the  light  was  a  nebulous 
ring  encircling  the  earth,  and  lying  within  the  orbit  of  the  moon. 

NOTE.  —  The  surface  of  the  sun  is  now  the  object  of  assiduous  observa- 
tion and  study.  The  shining  surface  of  the  sun,  whence  come  our  light 
and  heat,  is  called  the  photosphere.  Outside  of  this  is  the  chromosphere,  to 
which  the  red  flames  belong,  and  which  is  largely  composed  of  hydrogen 
and  the  vapors  of  metals.  Outside  of  the  chromosphere  lies  the  corona. 
Modern  research  has  disproved  much  that  was  formerly  believed  concern- 
ing the  physical  constitution  of  the  sun  and  its  envelope;  but  it  has  by  no 
means  established  what  that  constitution  really  is. 


SIDEREAL   AND   SOLAR   TIMES. 


CHAPTER  VII. 

SIDEREAL   AND    SOLAR   TIME.      THE    EQUATION    OF    TIME.      THE 
CALENDAR. 

104.  Sidereal  and    Solar   Days.  —  IT  is  important  to  distin- 
guish between  the  apparent  annual  motion  of  the  sun  in  the 
ecliptic,   from  west  to  east,  and  the  apparent  diurnal  motion 
towards  the  west,  which  the  rotation  of  the  earth  gives  to  all 
celestial  bodies  and  points.     A  sidereal  day  is  the  interval  of 
time  between  two  successive  transits  of  the  vernal  equinox  over 
the  same  branch  of  the  meridian.     A  solar  day  is  the  interval 
between  two  similar  transits  of  the  sun.     But  the  continuous 
motion  of  the  sun  towards  the  east  causes  it  to  appear  to  move 
more  slowly  towards  the  west  than  the  vernal  equinox  moves. 
The    solar  day  is   therefore  longer  than  the  sidereal  day,  the 
average  amount  of  the  difference  being  3m.  55.5s.     And  fur- 
thermore, in  the  interval  of  time  in  which  the  sun  makes  one 
complete  revolution  in  the  ecliptic,  the  number  of  daily  revo- 
lutions which  it  appears  to  make  about  the  earth  will  be  less 
by  one   than   the   number   of  daily  revolutions   made   by  the 
equinox.      The  sidereal  year,  then  (Art.   89),  which    contains 
865d.  6h.  9m.  9.6s.  of  solar  time,  contains  366d.   6h.  9m.  9.6s. 
of  sidereal  time.      [See  Note,  page  154.] 

105.  Relation  of  Sidereal  and  Solar  Times. — Since  the  side- 
real day  is  shorter  than  the  solar  day  (and,  consequently,  the 
sidereal  hour,  minute,  &c.,  than  the  solar  hour,  minute,  &c.),  it 
is  evident  that  any  given  interval  of  time  will  contain   more 
units  of  sidereal  than  of  solar  time.     The  relative  values  of  the 
sidereal  and  the  solar  days,  hours,  &c.,  are  obtained  as  follows: — 
We  have  from  the  prec2ding  article, 

366.25G36  sidereal  days  =  365.25636  solar  days: 

one  sidereal  day  =      (5.99727  solar  day, 
one  sidereal  hour  =      0.99727  solar  hour,  &c. 


EQUATION   OF   TIME.  101 

Having,  therefore,  an  interval  of  time  expressed  in  either  solar 
or  sidereal  units,  we  may  easily  express  the  same  interval  in 
units  of  the  other  denomination.  This  is  called  the  conversion 
of  a  solar  into  a  sidereal  interval,  and  the  reverse :  and  tables 
for  facilitating  this  conversion  are  given  in  the  Nautical  Al- 
manac. 

Again,  knowing  the  sidereal  time  at  any  instant,  the  hour- 
angle,  that  is  to  say,  of  the  vernal  equinox,  the  corresponding 
solar  time,  or  the  hour-angle  of  the  sun,  is  readily  obtained  by 
subtracting  from  the  sidereal  time  the  sun's  right  ascension. 
This  is  indeed  a  corollary  of  the  theorem  proved  in  Art.  9, 
from  which  we  see  that  the  sum  of  the  sun's  right  ascension 
(which  can  always  be  found  in  the  Nautical  Almanac),  and 
its  hour-angle,  is  the  sidereal  time.  Either  of  these  times,  then, 
may  be  converted  into  the  other.  \  "*  / 


THE   EQUATION   O 

106.  Inequality  of  Solar  Days. — Observation  shows  that  the 
length  of  the  solar  day  is  not  a  constant  quantity,  but  varies  at 
different  seasons  of  the  year,  and,  indeed,  from  day  to  day.  A 
distinction  must  therefore  be  made  between  the  apparent  or 
actual  solar  day,  and  the  mean  solar  day,  which  is  the  mean 
of  all  the  apparent  solar  days  of  the  year.  A  uniform  measure 
of  time  may  be  obtained  from  the  apparent  diurnal  motion 
with  reference  to  our  meridian  of  a  fixed  celestial  body  or 
point.  It  may  also  be  obtained  from  the  apparent  diurnal 
motion  of  a  celestial  body  which  changes  its  position  in  the 
heavens,  provided  that  two  conditions  are  satisfied ;  first,  the 
plane  in  which  the  body  moves  must  be  perpendicular  to  the 
plane  of  the  meridian:  and  second,  its  motion  in  that  plane 
must  be  uniform.  Both  these  conditions  are  so  very  nearly 
satisfied  by  the  motion  of  the  vernal  equinox,  that  any  two 
sidereal  days  may  be  considered  to  be  sensibly  equal  to  each 
other;  but  neither  condition  is  satisfied  by  the  motion  of  the 
sun.  It  moves  in  the  ecliptic,  the  plane  of  which  is  not,  in 
general,  perpendicular  to  the  plane  of  the  meridian :  and  its 
motion  in  this  plane  is  not  uniform.  We  have,  therefore,  two 


102  EQUATION    OF    TIME. 

causes  of  the  inequality  of  the  soltir  days,  the  effect  of  each  of 
which  we  will  now  proceed  to  examine. 

107.  Irregular  Advance  of  the  Sun  in  the  Ecliptic. — Observa- 
tion shows  that  the  sun's  motion  in  longitude  is  not  uniform. 
The  mean  daily  motion  is,  of  course,  obtained  by  dividing  360° 
by  the  number  of  days  and  parts  of  a  day  in  a  year,  and  is 
59'  8."3.  But  the  daily  motion  about  the  first  of  January  is 
61'  10",  while  about  the  first  of  July  it  is  only 
57'  12".  In  Fig.  42,  let  the  circle  AM'M" 
represent  the  apparent  orbit  of  the  sun  in 
the  ecliptic  about  the  earth  E,  and  let  the 
sun  be  supposed  to  be  at  the  point  A  where 
its  daily  motion  is  the  greatest,  on  the  first 
of  January.  Let  us  also  suppose  a  fictitious 
^si^n  (which  we  will  call  the  first  mean  sun) 
to  niqye  in  the  ecliptic  with  the  uniform  rate  of  59'  8". 3  daily, 
iik.H'l  t;>  )>o  ul,  the.  jximj.4  a^  tne  sanie  time  that  the  true  sun  is 
there.  On  the  next  day  the  mean  sun  will  have  moved  eastward 
to  some  point  M,  while  tli3  true  sun,  whose  daily  motion  is  at  this 
time  greater  than  that  of  the  mean  sun,  will  be  found  at  some 
point  T,  to  the  east  of  M.  The  true  sun  will  continue  to  gain  on 
the  mean  sun  for  about  three  months,  at  the  end  of  which  time 
the  mean  sun  will  begin  to  gain  on  the  true  sun,  and  will  finally 
overtake  it  at  the  point  B,  on  the  first  of  July.  During  the 
second  half  of  the  year  the  mean  sun  will  be  to  the  east  of  the 
true  sun,  and  at  the  end  of  the  year  the  two  suns  will  again  be 
together  at  A. 

The  angular  distance  between  the  two  suns,  represented  in 
the  figure  by  the  angles  TEN,  T'EM',  &c.,  is  called  the  Equa- 
tion of  the  Centre.  It  is  evidently  additive  to  the  mean  longi- 
tude of  the  sun  while  it  is  moving  from  A  to  B,  and  subtractive 
from  it  while  it  is  moving  from  B  to  A.  Its  greatest  value  is 
about  8  minutes  of  time. 

Since  the  rotation  of  the  earth  gives  to  both  these  bodies  a 
common  daily  motion  to  the  west,  it  is  plain  that  from  January 
to  July  the  mean  sun  will  cross  the  meridian  before  the  true 
sun,  and  that  from  July  to  January  the  true  sun  will  cross  the 
meridian  before  the  mean  sun. 


EQUATION   OF    TIME.  103 

108.  Obliquity  of  trie  Ecliptic  to  the  Meridian. — Even  if  the 
sun's  motion  in  the  ecliptic  were  uniform,  equal  advances  of  the 
sun  in  longitude  would  not  be  accompanied  by  equal  advances 
in  right  ascension,  in  consequence  of  the  obliquity  of  the 
ecliptic  to  the  meridian.  The  truth  of  this  may  be  seen  in 
Fig.  43.  Let  this  figure  represent 
the  projection  of  the  celestial  sphere 
on  the  plane  of  the  equinoctial  co- 
lure  PApH.  A  and  H  are  the 
equinoxes,  P  and  p  the  celestial 
poles,  AeH  the  equinoctial,  and 
A  EH  the  ecliptic.  Let  the  ecliptic 
be  divided  into  equal  arcs,  AB,  BC, 
&c.,  and  through  the  points  of  divi- 
sion, B,  C,  &c.,  let  hour-circles  be 
drawn,  meeting  the  equinoctial  in 
the  points  b,  c,  &c.  Now,  since  all  great  circles  bisect  each 
other,  AEH  is  equal  to  AeH,  and  if  Pep  is  the  projection  of  an 
hour-circle  perpendicular  to  the  circle  PApH,  AE  and  Ae  are 
quadrants,  and  equal.  The  angle  PBC  is  evidently  greater 
than  PAB,  PCD  is  greater  than  PBC,  &c. :  in  other  words,  the 
equal  arcs  AB,  BC,  &c.,  are  differently  inclined  to  the  equi- 
noctial. The  effect  of  this  is  that  the  equinoctial  is  divided 
into  unequal  parts  by  the  hour-circles  Pb,  PC,  &c.,  be  being 
greater  than  Ab,  cd  than  be,  &c.  It  is  to  be  noticed,  further, 
that  the  points  B  and  b,  being  on  the  same  hour-circle,  will  be 
on  the  meridian  at  the  same  instant  of  time:  and  the  same  is 
true  of  C  and  c,  D  and  d,  &c. 

Now,  if  A  is  the  vernal  equinox,  the  first  mean  sun,  moving 
in  the  ecliptic  with  the  constant  daily  rate  of  59'  8". 3,  will 
pass  through  that  point  on  the  21st  of  March.  Let  another 
fictitious  sun  (called  the  second  mean  sun)  leave  the  point  A  at 
the  same  time,  and  move  in  the  equinoctial  with  the  same  uni- 
form daily  rate.  Since  BAb  is  a  right-angled  triangle,  Ab  is 
less  than  AB.  Hence,  when  the  first  mean  sun  reaches  B,  the 
second  mean  sun  will  be  at  some  point  m,  to  the  east  of  b: 
when  the  first  mean  sun  is  at  C,  the  second  mean  sun  will  be  to 
the  east  of  c,  &c. :  and  the  second  mean  sun  will  continue  to 


104  EQUATION    OF    TIME. 

lie  to  the  east  of  the  first  mean  sun  until  the  21st  of  June  (the 
summer  solstice),  when  both  suns  will  be  at  the  points  E  and  e 
at  the  same  instant  of  time,  and  will  therefore  come  to  the 
meridian  together.  In  the  second  quadrant,  the  second  mean 
sun  will  lie  to  the  west  of  the  first  mean  sun,  and  both  suns 
will  reach  H,  the  autumnal  equinox,  on  the  21st  of  September. 
The  relative  positions  in  the  third  and  the  fourth  quadrant  will 
be  identical  with  those  in  the  first  and  the  second. 

From  the  21st  of  March,  then,  to  the  21st  of  June,  the  second 
mean  sun,  being  to  the  east  of  the  first  mean  sun,  will  come 
later  to  the  meridian ;  and  the  same  will  also  be  true  from  the 
21st  of  September  to  the  21st  of  December.  In  the  two  other 
similar  periods  the  case  will  be  reversed,  and  the  second  mean 
sun  will  come  earlier  to  the  meridian  than  the  first  mean  sun. 
The  greatest  difference  of  the  hour-angles  of  these  two  mean  suns 
is  about  10  minutes  of  time. 

109.  Equation  of  Time. — It  is  by  means  of  these  two  fictitious 
suns  that  we  are  able  to  obtain  a  uniform  measure  of  time  from 
the  irregular  advance  of  the  sun  in  the  ecliptic.  The  second 
mean  sun  satisfies  the  two  conditions  stated  in  Art.  106,  and 
therefore  its  hour-angle  is  perfectly  uniform  in  its  increase. 
This  hour-angle  is  the  mean  solar  time  of  our  ordinary  watches 
and  clocks.  The  hour-angle  of  the  true  sun  is  called  the  ap- 
parent solar  time:  and  the  difference  at  any  instant  between  the 
apparent  and  the  mean  solar  time  is  called  the  equation  of  time. 

Let  Fig.  44  be  a  projection  of  the 
celestial  sphere  on  the  plane  of  the 
horizon.  Z  is  the  zenith,  P  the 
pole,  EVQ  the  equinoctial,  CL  the 
ecliptic,  and  Fthe  vernal  equinox. 
Let  T  be  the  position  of  the  true 
sun  in  the  ecliptic,  and  M  that  of 
second  mean  sun  in  the  equinoctial. 
The  angle  TPM  is  evidently  the 
equatioi  of  time.  This  angle  is  mea- 
sured by  the  arc  AM,  or  VM—  VA : 

the  difference,  that  is,  of  the  right  ascensions  of  the  true  and 
tbe  second  mean  sun.  But  since  the  angular  advance  of  the 


JALEKDAR.  105 

second  mean  sun  in  the  equinoctial  is,  by  hypothesis,  as  shown 
in  the  previous  article,  equal  to  the  angular  advance  of  the  first 
mean  sun  in  the  ecliptic,  it  follows  that  the  right  ascension  of 
the  second  mean  sun  is  always  equal  to  the  longitude  of  the 
first  mean  sun,  or,  as  it  is  usually  called,  the  true  sun's  mean 
longitude.  The  equation  of  time,  then,  is  the  difference  of  the 
sun's  true  right  ascension  and  mean  longitude;  and  thus  com- 
puted is  given  in  the  Nautical  Almanac  for  each  day  in  the 
year.  It  reduces  to  zero  four  times  in  the  year,  and  passes 
through  four  maxima,  ranging  in  value  from  4  minutes  to  16 
minutes. 

110.  Astronomical  and  Civil  Time. — The  mean  solar  day  is 
considered  by  astronomers  to  begin  at  mean  noon,  when  the 
second  mean  sun  (usually  called  simply  the  mean  sun)  is  at  its 
upper  culmination.     The  hours  are  reckoned  from  Oh.  to  24h. 
The  mean  solar  day,  so  considered,  is  called  the  astronomical  day. 

The  civil  day  begins  at  midnight,  twelve  hours  before  the 
astronomical  day,  and  is  divided  into  two  parts  of  twelve  hours 
each,  called  A.M.  and  P.M. 

We  must,  therefore,  carefully  distinguish  between  any  given 
civil  time  and  the  corresponding  astronomical  time.  For  in- 
stance, January  3d,  8  A.M.,  in  civil  time,  is  the  same  as  Janu- 
ary 2d,  20h.,  in  astronomical  time. 

THE   CALENDAR. 

111.  Owing  to  causes  which  will  be  explained  further  on,  the 
position  of  the  vernal  equinox  is  not  absolutely  stationary,  but 
moves  westward  along  the  ecliptic,  with  an  annual  rate  of  about 
50". 2.     The  sun,  then,  moving  eastward  from  the  equinox,  will 
reach  it  again  before  it  has  made  one  complete  sidereal  revo- 
lution about  the  earth.     This  interval  of  time  in  which  the  sun 
moves  from  and  returns  to  the  equinox  is  called  a  tropical  year, 
and   consists  of  365d.  5h.   48m.  47.8s.     The   Julian  Calendar 
was  established  by  Julius  Csesar,  44  B.C.,  and  by  it  one  day  was 
inserted  in  every  fourth  year.     This  was  the  same  thing  as  as- 
suming that  the  length   of  the   tropical  year  was   365d.  6h., 
instead  of  the  value  given  above,  thus  introducing  an  accumu- 
lative  error    of    llm.    12s.    overy   year.      This    calendar   was 


106  CALENDAR. 

adopted  by  the  Church  in  325  A.p.,  at  the  Council  of  Nice,  and 
the  vernal  e  juinox  then  fell  on  the  21st  of  March.  In  1582, 
the  annual  error  of  llm.  12s.  caused  the  venial  equinox  to  fall 
on  the  llth  of  March,  instead  of  the  21st.  Pope  Gregory  XIII. 
therefore  ordered  that  ten  days  should  be  omitted  from  the  year 
1582,  and  thus  brought  the  vernal  equinox  back  again  to  the 
21st  of  March.  Furthermore,  since  the  error  of  llm.  12s.  a  year 
amounted  to  very  nearly  three  days  in  400  years,  it  was  decided 
to  leave  out  three  of  the  inserted  days  (called  intercalary  days) 
every  400  years,  and  to  make  this  omission  in  those  years  which 
were  not  exactly  divisible  by  400.  Thus  of  the  years  1700, 
1800,  1900,  2000,  all  of  which  are  leap  years  according  to  the 
Julian  calendar,  only  the  last  is  a  leap  year  according  to  the 
reformed  or  Gregorian  calendar.  By  this  calendar  the  annual 
error  is  only  24  seconds,  and  will  not  amount  to  a  day  in  much 
less  than  4000  years. 

This  reformed  calendar  was  not  adopted  by  England  until 
1752,  when  eleven  days  were  omitted  from  the  calendar.  The 
two  calendars  are  now  often  called  the  old  style  and  the  new  style. 
For  instance,  April  26th,  O.S.,  is  the  same  as  May  8th,  N.S.  In 
Russia  the  old  style  is  still  retained,  though  it  is  customary  to 
give  both  dates ;  as  1868,  ^?.  All  other  Christian  countries 
have  adopted  the  new  style. 


UNIVERSAL   GRAVITATION.  107 


CHAPTER  VIII. 

LAW  OF  UNIVERSAL  GRAVITATION.    PERTURBATIONS  IN  THE 
EARTH'S  ORBIT.    ABERRATION. 

112.  The  Law  of  Universal  Gravitation. — The  earth,  as  we 
have  seen  in  Chapter  VL,  revolves  about  the  sun  in  an  elliptical 
orbit,  with  a  linear  velocity  of  eighteen  miles  a  second.  At  every 
point  of  its  orbit  the  centrifugal  force  induced  by  this  revolution 
must  create  in  the  earth  a  tendency  to  leave  its  orbit,  and  to  go 
off  in  the  direction  of  a  tangent  to  the  orbit  at  that  point.  To 
counteract  this  centrifugal  force,  there  must  constantly  exist  a 
centripetal  force,  by  which  the  earth  is  at  every  instant  deflected 
from  this  rectilinear  path  which  it  tends  to  follow,  and  is  drawn 
towards  the  sun ;  and  in  order  that  the  orbit  of  the  earth  may 
remain  unchanged  in  form, — as  observation  shows  that  it  does 
remain, — these  two  forces  must  be  in  constant  equilibrium.  Ad- 
mitting, then,  the  existence  of  such  a  centripetal  force,  it  remains 
to  determine  the  nature  of  the  force,  and  the  laws  under  which 
it  acts. 

The  force  is  believed  to  be  identical  in  nature  with  that  force 
which  causes  all  bodies,  free  to  move,  to  tend  towards  the  earth's 
centre,  and  which  we  call  the  force  of  gravity.  At  whatever 
height  above  the  surface  of  the  earth  the  experiment  may  be 
made,  this  attractive  force  of  the  earth  is  found  to  exist ;  and 
there  is  no  good  reason  for  assuming  any  finite  limit  beyond 
which  this  force,  however  much  its  effects  may  be  lessened  by 
other  and  opposing  forces,  does  not  have  at  least  a  theoretic 
existence.  And,  furthermore,  as  the  sun  and  the  other  heavenly 
bodies  are  all  masses  of  matter  like  the  earth,  there  is  every 
reason  for  concluding  that  they  too,  as  well  as  the  earth,  possess 
this  power  of  attracting  other  bodies  towards  their  centres.  Nor 
is  this  attractive  power  a  characteristic  of  large  bodies  alone  :  for 


108  UNIVERSAL    GRAVITATION. 

we  have  already  seen  in  the  experiment  with  the  torsion  balance, 
described  in  Art.  66,  that  small  globes  of  lead  exert  a  sensible 
attraction  upon  still  smaller  globes.  We  may  therefore  assume 
that  what  is  true  of  each  of  these  masses,  large  and  small,  as  a 
whole,  is  no  less  true  of  the  separate  particles  of  which  it  is  com- 
posed, and  that  every  particle  of  matter  in  the  universe  has  an 
attractive  power  upon  every  other  particle. 

In  order  to  determine  the  laws  under  which  this  attractive 
power  is  exerted,  we  have  only  to  assume  that  the  laws  which 
are  shown  by  experiment  to  obtain  at  the  earth's  surface  hold 
equally  good  throughout  the  universe;  so  that  whatever  the 
masses  of  bodies  may  be,  or  whatever  the  distances  by  which 
they  are  separated  from  each  other,  the  forces  with  which  any 
two  bodies  attract  a  third  will  be  directly  proportional  to  the 
masses  of  the  two  attracting  bodies,  and  inversely  proportional 
to  the  squares  of  their  distances  from  the  third  body. 

This,  then,  is  Newton's  Law  of  Universal  Gravitation.  Every 
particle  of  matter  in  the  universe  attracts  every  other  particle, 
ivith  a  force  directly  proportional  to  the  mass  of  the  attracting 
particle,  and  inversely  proportional  to  the  square  of  the  distance 
between  the  particles.  In  applying  this  general  law  to  the  par- 
ticles which  compose  the  masses  of  the  heavenly  bodies,  Newton 
has  demonstrated  that  the  attraction  exerted  by  a  sphere  is  pre- 
cisely what  it  would  be  if  all  the  particles  in  the  sphere  were 
collected  at  its  centre,  and  constituted  one  particle,  with  an  at- 
tractive power  equal  to  the  sum  of  the  powers  of  these  different 
particles. 

113.  Verification  of  the  Law  in  the  Case  of  the  Moon.  —  The 
moon  is  shown  by  observation  to  revolve  about  the  earth  in  a 
period  of  27.32  days,  at  a  mean  distance  from  the  earth  of  238,800 
miles.  If  we  take  the  formula  for  centrifugal  force  given  in 
Art.  69, 


and  substitute  for  r  the  moon's  distance  in  feet,  and  for  t  its 
period  of  revolution  in  seconds,  we  shall  find  for  the  centrifugal 
force, 

/=  0.0089  feet: 


) 

MASS   OF   THE   SUN.  109 

that  is  to  say,  in  one  second  the  earth  tends  to  give  the  moon  a 
velocity  towards  itself  of  0.0089  feet.  Now  the  force  of  gravity 
on  the  eartn's  surface  at  the  equator  is  32.09  feet;  and  if  the  law 
of  gravitation  is  assumed  to  be  true,  the  force  of  gravity  at  the 

32  09 

distance  of  the  moon  will  be  -     — —  feet,  since  the  distance  of 

(oO.Zo7) 

the  moon  from  the  earth  is  equal  to  60.267  of  the  earth's  radii. 
The  value  of  this  expression  is  found  to  be  0.0088  feet.  The  two 
results  vary  by  only  joio^h  of  a  foot:  and  it  is  therefore  fair  to 
conclude  that  the  centrifugal  force  of  the  moon  in  its  orbit  is 
really  counteracted  by  the  earth's  attraction. 

In  whatever  way  the  law  of  gravitation  is  tested  in  connection 
with  the  observed  motions  of  the  heavenly  bodies,  the  facts  which 
come  by  observation  are  always  found  to  be  in  close  agreement 
with  the  results  which  the  law  demands ;  and  it  is  safe  to  say 
that  the  truth  of  this  law  is  as  satisfactorily  demonstrated  as  is 
that  of  the  laws  of  refraction,  of  the  laws  of  sound,  or  of  the 
many  other  natural  laws  which  depend  upon  observation  and 
experiment  for  their  ultimate  proof. 

114.  The  Mass  of  the  Sun. — Let  A  denote  the  attraction  ex- 
erted by  the  sun  on  the  earth,  and  a  that  exerted  by  the  earth 
on  a  body  at  its  surface.  Let  If  denote  the  mass  of  the  sun,  m 
that  of  the  earth,  r  the  radius  of  the  earth,  and  R  the  radius 
of  the  earth's  orbit.  We  have,  then,  by  the  law  of  gravitation, 

A M        r2 

a        m        R2 
But  A  must  equal  the  earth's  centrifugal  force  in  its  orbit,  or 

.8    »  in  which  t  is  365.256  days,  reduced  to  seconds,  and  R  is 

expressed  in  feet.  We  have  also  a  equal  to  32.09  feet.  Substi- 
tuting these  values,  we  have, 


m  ~  32.09f  r2 

Substituting  the  known  values  of  the  different  quantities,  we 
shall  have 

—  =  327,000: 
m 

or  the-  mass  of  the  sun  is  equal  to  that  of  327,000  earths. 
Jo 


110  MOTION    OF   THE    EARTH 

The.  density  of  the  sun,  compared  with  that  of  the  earth,  being 

327,000 
equal  to  the  mass  divided  by  the  volume,  is  r»o 


density  is  therefore  equal  to  about  Hh  of  that  of  the  earth,  and 
to  about  f  ths  of  that  of  water. 

115.  Weight  of  Bodies  at  the  Surface  of  the  Sun.—  The  weight 
of  the  same  body  at  the  surface  of  the  earth  and  at  that  of  the 
sun  will  be  directly  as  the  masses  of  the  two  spheres  and  inversely 
as  the  squares  of  their  radii.     We   shall  find  that  the  weight 
of  a  body  at  the  sun  is  about  27.7  times  its  weight  at  the  earth; 
so  that  a  body  which  exerts  a  pressure  of  10  pounds  at  the  earth 
would  exert  a  pressure  at  the  sun  equal  to  that  of  277  of  the 
same  pounds;  and  a  man  whose  weight  is  150  pounds  would, 
if  transported  to  the  sun,  be  obliged  to  support  in  his  own  body 
a  weight  equivalent  to  about  two  of  our  tons. 

116.  The  Earth's  Motion  at  Perihelion   and  Aphelion.  —  We 
have  already  seen  that  the  angular  velocity  of  the  earth  in  its 
orbit  is  the  greatest  at  perihelion,  when  the  earth  is  the  nearest  to 
the  sun,  and  is  the  least  at  aphelion,  when  the  earth  is  the  farthest 
from  the  sun.     This  irregularity  of  motion  is  a  consequence  of 
the  attraction  exerted  by  the  sun  on  the  earth,  as  may  be  seen  in 

Fig.  45.  In  this  figure,  let  S  be  the  sun, 
P  the  perihelion  of  the  earth's  orbit,  and 
A  the  aphelion.  Let  the  earth  move 
from  A  to  P,  and  suppose  it  to  be  at  the 
point  E.  The  attraction  of  the  sun  on 
the  earth,  along  the  line  ES,  may  be  re- 
solved into  two  forces,  one  of  which, 
acting  in  the  direction  of  the  tangent  EB, 
B  will  evidently  tend  to  increase  the  velo- 

Fig.  45.  city  of  the  earth  in  its  orbit.     At  the 

point  E",  on  the  contrary,  where  the  earth  is  moving  toward  A, 
the  effect  of  the  sun?s  attraction  is  to  diminish  the  earth's  velocity. 
lu  general,  then,  the  earth's  velocity  will  increase  as  it  moves 
from  aphelion  to  perihelion,  and  decrease  as  it  moves  from  peri- 
helion to  aphelion.  • 


KEPLER'S  LAWS.  Ill 


KEPLER  S   LAWS. 

117.  In  the  early  part  of  the  seventeenth  century,  more  than 
tifty  years  before  the  announcement  by  Newton  of  the  law  of 
universal  gravitation,  the  astronomer  Kepler,  by  an  examination 
of  the  observations  which  had  been  made  upon  the  motions  of  the 
planets,  and  which  had  shown  that  the  planets  revolved  about 
the  sun,  discovered  the  following  laws : — 

(1.)  The  orbit  of  every  planet  is  an  ellipse,  having  the  sun  at 
one  of  its  foci. 

(2.)  If  a  line,  called  a  radius  vector,  is  supposed  to  be  drawn 
from  the  sun  to  any  planet,  the  areas  described  by  this  line,  as 
the  planet  revolves  in  its  orbit,  are  proportional  to  the  times. 

(3.)  The  squares  of  the  times  of  revolution  of  any  two  planets 
are  proportional  to  the  cubes  of  their  mean  distances  from  the  sun. 

These  laws  were  verified  by  Newton  in  his  Principia,  in  a 
course  of  mathematical  reasoning,  the  foundation  of  which  was 
his  theory  of  universal  gravitation.  With  regard  to  Kepler's 
first  law,  he  showed  that  any  two  spherical  bodies,  mutually  at- 
tracted, describe  orbits  about  their  common  centre  of  gravity, 
and  that  these  orbits  are  limited  in  form  to  one  or  another  of  the 
four  conic  sections, — the  circle,  the  ellipse,  the  parabola,  and  the 
hyperbola.  For  instance,  in  the  case  of  the  earth  and  the  sun, 
the  earth  does  not  describe  an  ellipse  about  the  sun  at  rest,  but 
both  earth  and  sun  revolve  about  their  common  centre  of  gravity. 
It  is  shown  in  Mechanics  that  the  common  centre  of  gravity  of 
any  two  globes  is  at  a  point  on  the  straight  line  joining  their  in- 
dependent centres  of  gravity,  so  situated  that  its  distances  from 
the  centres  of  the  two  globes  are  inversely  as  the  masses  of  the 
globes.  Hence  the  distance  of  the  common  centre  of  gravity 

of  the  earth  and  the  sun  from  the  centre  of  the  sun  is      '       '  — 

oJ7,U()0 

miles,  or  only  about  280  miles ;  so  that  the  sun  may  practically 
bo  considered  to  be  at  rest. 

With  regard  to  Kepler's  second  law,  Newton  further  showed 
that  the  angular  velocity  with  which  the  radius  vector  moves 
must  be  inversely  proportional  to  the  square  of  its  length. 


112  PRECESSION. 

Finally,  he  proved  also  the  tru^h  of  the  third  law,  provided 
only  that  a  slight  correction  is  introduced  when  the  mass  of  the 
planet  is  not  so  small  as  to  be  inappreciable  in  comparison  with 
that  of  the  sun.  If  t  and  t'  denote  the  times  of  revolution  of 
any  'two  planets,  m  and  m'  their  masses  (the  mass  of  the  sun 
being  unity),  and  d  and  d'  their  mean  distances  from  the  sun, 
Newton  showed  that  we  shall  always  have  the  following  pro- 
portion : 


I  +  m'  1  +  m' 

If  m  and  m'  are  so  small  that  they  may  be  omitted  without 
sensible  error,  this  proportion  becomes  identical  with  Kepler's 
third  law. 


PERTURBATIONS  IN  THE  EARTHS  ORBIT. 


118.  Precession. — Although  the  absolute  positions  of  the 
planes  of  the  ecliptic  and  the  equinoctial  in  space,  and  their  re- 
lative positions  to  each  other,  remain  very  nearly  the  same  from 
year  to  year,  there  are  nevertheless  certain  small  perturbations 
in  these  positions  which  are  made  evident  to  us  by  refined  and 
extended  observations.  The  principal  of  these  perturbations  is 
called  precession. 

The  latitudes  of  all  the  fixed  stars  remain  very  nearly  the 
same  from  year  to  year,  and  even  from  century  to  century:  and 
we  therefore  conclude  that  the  position  of  the  ecliptic  with 
reference  to  the  celestial  sphere  remains  very  nearly  unchanged. 
But  the  longitudes  of  the  stars  are  all  found  to  increase  by  an 
annual  amount  of  50".2:  and  hence  the  line  of  the  equinoxes 
must  have  an  annual  westward  motion  of  the  same  amount. 
This  westward  motion  is  called  the  precession  of  the  equinoxes. 
Since  the  ecliptic  remains  stationary  in  the  heavens  (or  at  least 
so  nearly  stationary  that  the  latitudes  of  the  stars  only  vary 
by  half  a  second  of  arc  in  a  year),  this  precession  must  be  con- 
sidered to  be  a  motion  of  the  equinoctial  on  the  ecliptic. 

In  Fig.  46,  let  EQ  represent  the  equinoctial,  and  LC  the  eclip- 
tic. .BFis  the  line  of  the  equinoxes,  which  moves  about  in  the 
plane  of  the  ecliptic,  taking  in  course  of  time  the  new  position 
B'V.  Perhaps  the  clearest  conception  of  this  motion  is  ob- 


PRECESSION. 


113 


tained  by  considering  P,  the  pole  of 
the  equinoctial,  to  revolve  about  A, 
the  pole  of  the  ecliptic,  in  the  circle 
PG  (the  polar  radius  of  which,  AP, 
is  23°  27'),  moving  westward  in  this 
circle  with  the  annual  rate  of  50".2, 
and  completing  its  revolution  in 
25,817  years. 

A  general  explanation  of  the 
cause  of  precession  may  be  given  by 
means  of  Fig.  47.  The  earth  may 
be  regarded  as  a  sphere  surrounded  by  a  spheroidal  shell,  as 
represented  in  the 
figure  by  EPQp, 
and  the  matter  in 
this  shell  may  be 
considered  to  form 
a  ring  about  the 
earth  in  the  plane 
of  the  equator,  as 
shown  in  E'FQ'p'. 
It  is  to  the  attrac- 
tion of  the  sun  and 
the  moon  on  this 
ring,  combined  with 
the  earth's  rotation, 
that  the  precession 
of  the  equinoxes  is 
due.  In  the  figure 
let  S  be  the  sun,  the 
circle  ABV  the  ecliptic,  and  E"Q"  this  equatorial  ring  of  the 
earth.  Let  J.CTbe  the  plane  of  the  equinoctial,  meeting  the 
plane  of  the  ecliptic  in  the  line  of  equinoxes  A  V.  This  plane 
ij>  by  definition  determined  at  each  instant  by  the  position  of 
the  earth's  equator.  The  attraction  exerted  by  the  sun  upon 
the  different  particles  of  the  ring  in  that  half  of  it  which  is 
nearer  the  sun  (the  particle  E",  for  instance),  may  be  resolved 
into  two  forces,  one  acting  in  the  plane  of  the  equator,  and  the 


Fig.  47. 


114  PRECESSION. 

other  in  a  direction  perpendicular  4;o  that  plane,  or  in  the  direc- 
tion E"d.  The  sun's  attraction  upon  the  nearer  half  of  the 
ring,  then,  tends  to  draw  the  plane  of  the  ring  nearer  to  the 
plane  of  the  ecliptic.  On  the  other  hand,  the  sun's  attraction 
upon  the  farther  half  of  the  ring  tends  to  bring  about  the 
opposite  result ;  but  since,  by  the  law  of  attraction,  the  latter 
effect  is  less  than  the  former,  we  may  consider  the  whole  result 
of  the  sun's  attraction  upon  the  ring  to  be  a  tendency  in  the 
plane  of  the  ring  to  come  nearer  to  the  plane  of  the  ecliptic. 
Therefore,  if  the  ring  did  not  rotate,  the  plane  of  the  earth's 
equator  would  ultimately  come  into  coincidence  with  the  plane 
of  the  ecliptic. 

But  the  ring  does  rotate,  about  an  axis  perpendicular  to  its 
own  plane ;  and  the  combined  result  of  this  rotation  and  of  the 
rotation  about  the  line  of  the  equinoxes,  above  described,  is 
that  the  plane  of  the  equinoctial,  while  it  preserves  constantly 
its  inclination  to  the  plane  of  the  ecliptic,  moves  about  in  a 
westerly  direction:  the  line  of  intersection  of  the  two  planes 
also  moving  about  in  the  same  direction,  and  thus  giving  rise  to 
the  precession  of  the  equinoxes. 

Similar  results  will  evidently  follow  if  S  represents  the  moon 
instead  of  the  sun.  Owing  to  the  greater  proximity  of  the 
moon  to  the  earth,  however,  the  results  of  its  attraction  are 
more  than  double  those  of  the  attraction  of  the  sun.  There  is 
still  another  perturbation  in  the  position  of  the  line  of  the  equi- 
noxes which  is  a  result  of  the  mutual  attraction  between  the 
earth  and  the  other  planets.  This  attraction  tends  to  draw  the 
earth  out  of  the  plane  of  the  ecliptic,  without  affecting  in  any 
way  the  position  of  the  plane  of  the  equinoctial.  The  result  is 
an  annual  movement  of  the  equinoxes  towards  the  east.  This 
perturbation  is  exceedingly  minute,  being  only  about  yth  of  a 
second  of  arc  in  a  year.  The  value  50". 2  is  the  algebraic  sum 
of  all  these  perturbations. 

119.  Results  of  Precession.  —  One  result  of  precession  is  to 
make  the  interval  of  time  between  two  successive  returns  of  the 
sun  to  the  vernal  equinox  less  than  the  time  of  one  sidereal 
revolution,  by  the  time  required  by  the  sun  to  pass  over  50". 2, 
which  is  20m.  21.8s.  Hence  we  have  the  tropical  year,  to  which 


NUTATION.  115 

reference  has  already  been  made  in  Art.  111.  Another  result 
is  that  the  signs  of  the  Zodiac  (Art.  91)  no  longer  coincide  with 
the  constellations  after  which  they  are  named,  but  have  retreated 
towards  the  west  by  about  28°,  or  nearly  one  sign :  so  that  the 
constellation  of  Aries  now  lies  in  the  sign  of  Taurus.  Still 
another  result  is  that  the  same  star  is  not  the  pole-star  in  dif- 
ferent ages.  Referring  to  Fig.  46,  the  pole  of  the  heavens,  P, 
will  have  revolved  about  A  to  the  position  G,  in  the  course  of 
about  13,000  years;  and  a  star  of  the  first  magnitude,  called 
Vega,  which  is  now  about  51°  from  the  pole,  will  at  that  time  be 
less  than  5°  from  the  pole,  and  will  be  the  pole-star. 

120.  Nutation. — Since  precession  is  the  result  of  the  tendency 
of  the  sun  to  change  the  position  of  the  plane  of  the  equator,  it 
is  evident  that  there  will  be  no  precession  when  the  sun  is  itself 
in  the  plane  of  the  equator, — in  other  words,  at  the  equinoxes, — 
and  that  the  precession  will  be  at  its  maximum  when  the  sun  is 
the  farthest  from  the  plane  of  the  equator:  that  is  to  say,  is  at  the 
solstices.     The  amount  of  precession  due  to  the  influence  of  the 
moon  is  subject,  to  a  similar  variation,  being  the  greatest  when 
the  moon's  declination  is  the  greatest.     The  result  is  that  the 
pole  of  the  heavens  has  a  small  oscillatory  motion  about  its 
mean  place.     This  motion  is  called  nutation.     If  the  effect  of 
nutation  could  be  separated  from  that  of  precession,  the  pole 
would  be  found  to  move  in  a  very  minute  ellipse,  having  a 
major  axis  of  18". 5,  and  a  minor  axis  of  13". 7,  the  period  of 
one  revolution  in  this  ellipse  being  about  nineteen  years.    Since, 
however,  these  two  perturbations  co-exist,  the  result  is  that  the 
pole  of  the  heavens  revolves  about  the  pole  of  the  ecliptic,  not 
in  a  circle,  but  in  an  undulating  curve,  as 

represented  in  Fig.  48 :  the  amount  of  the 
deviation  being  very  much  exaggerated  in 
the  figure. 

121.  Change  in  the  Obliquity  of  the  Eclip- 
tic.— It  was  stated  in  Art.  118  that  the  lati- 
tudes of  the  stars  were  found  to  vary  from 
year   to   year   by  a  very  minute  amount. 

This  change  in  the  latitudes  is  due  to  a  change  in  the  position 
of  the  plane  of  the  ecliptic,  involving  a  change  in  the  obliquity 


116  ABERRATION. 

of  the  ecliptic.  The  obliquity  .of  the  ecliptic  in  1878  was 
23°  27'  18":  and  the  annual  amount  of  diminution  to  which  it 
is  now  subject  is  0".46.  Mathematical  investigations  show  that 
after  certain  moderate  limits  have  been  reached  this  diminution 
will  cease,  and  the  obliquity  will  begin  to  increase.  The  arc 
through  which  the  obliquity  oscillates  is  about  1°  21',  and  the 
time  of  one  oscillation  is  about  ten  thousand  years. 

122.  Advance  of  the  Line  of  Apsides. — The  line  connecting 
the  earth's  perihelion  and  aphelion  is  called  the  line  of  ajisides. 
This  line  revolves  from  west  to  east,  with  an  annual  rate  of  11". 8; 
a  perturbation  due  to  the  attraction  exerted  on  the  earth  by  the 
superior  planets.    The  time  in  which  the  earth  moves  from  peri- 
helion to  perihelion  is  called  the  anomalistic  year  (from  anomaly, 
Art.  98).     It  is  evidently  longer  than  the  sidereal  year,  and  is 
found  to  contain  365d.  6h.  13m.  49.3s. 

ABERRATION. 

123.  The  apparent  direction  of  a  celestial  body  is  determined 
by  the  direction  of  the  telescope  through  which  it  is  observed. 
In  consequence  of  the  motion  of  the  earth,  and  the  progressive 
motion  of  light,  the  telescope  is  carried  to  a  new  position  while 
the  light  is  descending  "through  it,  and  therefore  the  apparent 
direction  of  the  body  will  differ  from  its  true  direction. 

\?  In  Fig.  49,  let  OF  be  the  posi- 

tion of  the  axis  of  a  telescope  at 
the  instant  when  the  rays  of  light 
from  the  star  S  reach  the  object 
glass  0.  The  rays,  after  passing 
through  the  glass,  begin  to  con- 
verge towards  a  fixed  point  in 
space,  with  which,  at  this  instant, 
the  intersection  of  the  cross-wires 
Fig.  49.  coincides.  Let  the  earth  be  moving 

in  the  direction  FA.  Since  the  transmission  of  light  is  not 
instantaneous,  time  is  required  for  the  light  to  pass  from  0  to 
the  fixed  point  in  space,  and  in  that  time  the  earth  will  carry 
the  axis  of  the  telescope  to  some  new  position  0'Ff.  The  cross- 
wires  will  then  be  at  F't  while  the  rays,  whose  motion  in  space 


ABERRATION.  117 

is  enfirely  independent  of  any  motion  of  the  telescope,  will  tend 
to  meet  at  the  point  F.  In  order,  then,  to  have  the  image  of  the 
star  coincide  with  the  intersection  of  the  wires,  the  telescope 
must  be  so  moved  that  its  axis  will  lie  in  the  position  O'F.  The 
star  will  then  appear  to  lie  in  the  direction  FSi,  while  its  true  di- 
rection is  of  course  FS:  and  the  angle  which  these  two  directions 
make  with  each  other,  or  the  angle  F'  O'F,  is  called  the  aberra- 
tion. Representing  this  angle  by  A,  and  the  angle  OFF' ,  the 
angle  between  the  apparent  direction  of  the  star  and  the  direc- 
tion in  which  the  earth  is  moving,  by  I,  we  shall  have, 

sin  4:  BinI=FFf:   O'F'. 

But  the  ratio  FF' :  O'F'  is  the  ratio  between  the  velocity  of 
the  earth  and  that  of  light :  so  that  the  sine  of  the  aberration  is 
equal  to  the  ratio  of  the  velocity  of  the  earth  to  that  of  light, 
multiplied  by  the  sine  of  the  angle  J. 

124.  Diurnal  Aberration. — Aberration    causes    the    celestial 
bodies  to  appear  to  be  nearer  than  they  really  are  to  that  point 
of  the  celestial  sphere  towards  which  the  motion  of  the  earth 
is  directed  at  the  instant  of  observation.     As  a  correction,  then, 
it  is  to  be  applied  in  the  opposite  direction.     There  is  evidently 
no  aberration  when  the  motion  of  the  earth  is  directly  towards 
the  star,  and  the  greatest  amount  of  aberration  occurs  when 
the  direction  of  the  earth's  motion  is  at  right  angles  to  the 
direction  of  the  star. 

Aberration  is  of  two  kinds,  corresponding  to  the  daily  and 
the  yearly  motion  of  the  earth.  The  diurnal  aberration  tends 
to  displace  all  bodies  in  the  direction  in  which  the  earth  is 
carrying  the  observer :  that  is  to  say,  in  an  easterly  direction. 
It  evidently  varies  -with  the  linear  velocity  of  the  observer,  and 
is  therefore  the  greatest  at  the  equator  and  zero  at  the  poles. 
Owing  to  the  minuteness  of  the  velocity  of  any  point  of  the 
earth's  surface  about  the  axis  in  comparison  with  the  velocity 
of  light,  the  diurnal  aberration  is  extremely  small,  its  greatest 
value  being  less  than  Jd  of  a  second  of  arc. 

125.  Annual  Aberration. — The  displacement  of  a  star  occa- 
sioned by  the  motion  of  the  earth  in  its  orbit  about  the  sun  is 
called  the  annual  aberration.     The  effect  which  it  has  on  the  mo- 
tion of  any  body  will  depend  on  the  relative  situation  of  that  body 


118 


ABERRATION. 


to  the  plane  of  the  ecliptic,  as  ma^r  be  seen  in  Fig  50.  In  thia 
figure  /S  represents  the  sun,  ABCD  the  orbit  of  the  earth,  and  K 
the  pole  of  the  ecliptic.  Suppose  a  star  to  be  at  K.  As  the  earth 
moves  through  A,  in  the  direction  indicated  by  the  arrow,  the 
star  will  be  displaced  from  Kiv  a;  as  the  earth  moves  through 

B,  the  star  will  be  seen 
at  b,  &c.  Since  the 
direction  of  this  star  is 
always  at  right  angles 
to  the  direction  in 
which  the  earth  is 
moving,  the  aberra- 
tion will  continually 
be  at  its  maximum, 
as  shown  in  the  pre- 
vious article,  and  the 
star  will  describe  a 
circle  about  its  true 
place  as  a  centre.  If 
the  star  is  in  the  plane 
•of  the  ecliptic,  as  at  s,  there  will  be  no  aberration  when  the 
earth  is  at  A  or  (7,  and  the  aberration  will  be  at  its  maximum 
when  the  earth  is  at  B  or  D.  The  star  will  therefore  during 
the  year  describe  the  arc  b'd',  equal  in  value  to  twice  the  maxi- 
mum of  aberration,  and  having  the  true  place  of  the  star  at  its 
middle  point.  If  the  star  is  situated  between  the  pole  and  the 
plane  of  the  ecliptic,  it  will  describe  an  ellipse,  the  semi-major 
axis  of  which  is  the  maximum  of  aberration,  and  the  semi- 
minor  axis  of  which  increases  with  the  latitude  of  the  star. 

126.  Velocity  of  Light. — The  maximum  value  of  aberration  is 
the  same  for  all  bodies,  and  may  be  obtained  by  observing  the 
apparent  motion  of  a  fixed  star  during  the  year.  Its  value 
has  thus  been  obtained,  and  is  20". 4.  Now,  since  the  maximum 
of  aberration  occurs  when  the  angle  J,  in  the  formula  in  Art. 
123,  is  90°,  we  shall  have,  denoting  this  maximum  by  A't 

•        A>  FF' 

sin  A  -  -  Q,p,- 
But  FF'  is  the  velocity  of  the  earth  in  its  orbit,  or  18.4  miles 


ABERRATION.  H9 

a  second.  Hence,  the  velocity  of  light  in  a  second  is  18.4 
miles  multiplied  by  the  cosecant  of  20".4,  which  will  be  found 
to  be  185,600  miles.  Experiments  of  a  totally  different  character 
have  given  almost  precisely  the  same  result ;  and  it  is  believed 
that  this  estimate  is  within  a  thousand  miles  of  the  true  velocity. 

If  we  divide  the  disr.uice  of  the  earth  from  the  sun  by  this 
velocity,  we  find  that  it  requires  8m.  18s.  for  light  to  pass  over 
that  distance.  When  we  look  at  the  sun,  therefore,  we  see  it, 
not  as  it  is  at  the  time  of  observation,  but  as  it  was  8m.  18s. 
previously;  and  in  the  same  way  every  other  celestial  body 
appears  to  be  in  a  different  position  from  that  which  it  really 
occupies  at  the  instant  we  observe  it.  It  may  be  well  to  notice 
here,  that  the  time  required  for  light  to  pass  from  any  celestial 
body  to  the  earth  is  an  element  in  the  computation  of  the  appa- 
rent place  of  that  body  given  in  the  Nautical  Almanac.  In 
the  case  of  a  body  which  changes  its  actual  position  on  the  celes- 
tial sphere,  as  a  planet,  for  instance,  allowance  must  also  be  made 
for  what  is  called  planetary  aberration ;  since  even  if  the  earth 
were  stationary,  the  apparent  position  of  the  body  would  be 
behind  its  true  position  by  the  amount  of  its  motion  in  the  time 
required  for  light  to  come  from  the  body  to  the  earth.* 

127.  Aberration  a  Proof  of  the  Earth's  Revolution  about  the 
Sun. — The  existence  of  the  phenomenon  of  aberration,  as  de- 
scribed in  Art..  125,  is  a  matter  of  undoubted  observation :  and 
when  the  close  agreement  of  the  velocity  of  light  obtained  in 
the  preceding  article  with  the  velocity  obtained  by  independent 
philosophical  experiments  is  taken  into  consideration,  it  is  fair 
to  regard  the  existence  of  aberration  as  a  strong  direct  proof  of 
the  revolution  of  the  earth  about  the  sun.  Another  proof,  simi- 
lar in  many  respects  to  this,  will  be  noticed  when  we  come  to 
the  subject  of  the  eclipses  of  Jupiter's  satellites. 

*  Herschel  suggests  (Outlines  of  Astronomy,  %  385)  that  this  might  be 
called  the  equation  of  light,  in  order  to  prevent  its  being  confounded  with  tho 
real  aberration  of  light. 


120  ORBIT   OF   THE   ITOCXN. 


CHAPTER  IX. 

THE   MOON. 

128.  The  Orbit  of  the  Moon. — While  the  moon,  in  common 
with  all  the  celestial  bodies,  has  the  apparent  westward  motion 
which  is  due  to  the  rotation  of  the  earth,  it  also  changes  its  rela- 
tive position  to  the  other  bodies,  and  is  continually  falling  behind, 
or  to  the  east  of  them,  in  this  diurnal  motion.  In  other  words, 
it  has  an  independent  motion,  either  real  or  apparent,  from  west 
to  east.  This  eastward  motion  is  so  rapid,  that  we  only  need  to 
observe  the  relative  situations  of  the  moon  and  some  conspicuous 
star,  during  a  few  hours  on  any  favorable  night,  to  notice  a  per- 
ceptible change  in  their  angular  distance.  If  the  right  ascension 
and  the  declination  of  the  moon  are  determined  from  day  to  day, 
precisely  as  the  same  elements  of  the  sun's  position  were  deter- 
mined (Art.  89),  and  the  corresponding  positions  are  laid  down 
upon  a  celestial  globe,  we  shall  find  that  the  moon  makes  a  com- 
plete revolution  in  the  heavens,  about  the  earth  as  a  centre,  in 
an  average  period  of  27d.  7h.  43m.  11.5s.  We  shall  also  find 
that  the  plane  of  the  moon's  orbit  intersects  the  plane  of  the 
ecliptic  at  an  angle  whose  mean  value  is  5°  8'  44",  and  in  a  line 
which,  like  the  earth's  line  of  equinoxes,  is  continually  revolving 
towards  the  west :  so  that  the  apparent  orbit  of  the  moon  is  not 
a  circle,  but  a  kind  of  spiral.  This  revolution  is  much  more 
rapid,  however,  in  the  case  of  the  moon,  the  amount  of  retro- 
gradation  being  about  1°  27'  in  a  month,  and  the  complete  revo- 
lution being  effected  in  18.6  years. 

The  movement  of  the  moon  in  its  orbit  is  represented  in  Fig. 
51.  Let  E  be  the  earth,  and  the  circle  MANC  the  plane  of  the 
ecliptic.  Let  the  moon  be  at  J/at  any  time.  Then  will  MN  be 
the  line  in  which  the  plane  of  the  moon's  orbit  intersects  the 
plane  of  the  ecliptic.  This  liiie  is  calk d  the  line  of  the  nodes.  That 


ORBIT    OF    THE    MOON.  121 

extremity  of  the  line  through  which  the  moon  passes  in  moving 
from  the  southern  to  the  northern 
side  of  the  ecliptic  is  called  the 
ascending  node,  the  other  the  de- 
scending node.  Let  the  moon 
move  on  from  M  in  the  arc  MB. 
When  it  descends  to  the  ecliptic, 
it  will  meet  it,  not  at  the  point  N, 
but  at  some  point  N'.  and  the  line 
of  the  nodes  will  take  the  new  posi- 
tion N'M'.  The  moon  moves  on  to  the  other  side  of  the  ecliptic, 
passes  through  the  arc  N'D,  and  when  it  again  returns  to  the 
ecliptic,  will  meet  it,  not  at  Mf,  but  at  some  point  M",  and  the 
line  of  the  nodes  will  take  the  position  M"N".  The  revolution 
of  the  line  of  the  nodes  is  evidently  in  an  opposite  direction  to 
that  in  which  the  moon  itself  revolves,  and  is  therefore  from 
east  to  west. 

129.  Cause  of  the  Retrogradation  of  the  Nodes. — This  retro- 
grade movement  of  the  moon's  nodes  is  similar  in  character  to 
the  precession  of  the  equinoxes,  and  is  due  to  the  attraction 
which  the  sun  exerts  upon  the  moon.     Since  the  plane  of  the 
moon's  orbit  is  inclined  to  the  plane  of  the  ecliptic,  the  attrac- 
tion of  the  sun  will,  in  general,  tend  to  draw7  the  moon  out  of  its 
orbit  towards  the  ecliptic.     The  only  exceptions  to  this  rule  will 
occur  when  the  moon  is  at  one  of  its  nodes,  and  is  therefore  i.i 
the  plane  of  the  ecliptic,  and  also  when  the  line  of  the  moon's 
nodes  passes  through  the  sun,  at  which  time  the  attraction  of  the 
sun  is  exerted  along  this  line,  and  consequently  in  the  plane  of 
the  moon's  orbit.     In  Fig.  51  let  the  moon  be  at  B,  and  the  sun 
anywhere  in  the  ecliptic  except  on  the  line  of  the  nodes.     As  the 
moon  moves  on,  the  sun  is  continually  drawing  it  down  to  the 
ecliptic,  and  it  will  hence  meet  the  ecliptic,  not  at  N,  but  at  jY'. 
The  same  effect  will  be  seen  at  every  other  position  of  the  moon 
in  its  orbit,  with  only  the  exceptions  already  mentioned. 

130.  Change  in  the  Obliquity  of  the  Moon's  Orbit. — It  is  evi- 
dent from  the  same  figure  that  the  angle  which  the  arc  BN' 
makes  with  the  plane  of  the  ecliptic  is  greater  than  the  angle 

which  an  arc  drawn  through  B  and  N  would  make.     The  obli- 
11 


122  ORBIT    OF   THE   MOOX. 

quity  of  the  plane  of  the  moon's  orbit  is  therefore  increased  a* 
the  moon  approaches  the  node.  It  may  be  shown  in  the  same  way 
that  the  obliquity  is  diminished  as  the  moon  recedes  from  the  node. 
The  extreme  limits  which  this  angle  attains  are  5°  20'  6"  and 
4°  57'  22". 

131.  Elliptical  form  of  the  Moon's  Orbit. — The  angular  dia- 
meter of  the  moon  varies  at  different  points  of  its  orbit,  while 
its  mean  value  remains  the  same  from  month  to  month ;  we  there- 
fore conclude,  as  we  concluded  in  the  case  of  the  sun,  that  its 
distance  from  the  earth  is  not  constant,  the  greatest  distance  cor- 
responding to  the  least  diameter,  and  the  least  distance  to  the 
greatest   diameter.     If  we  neglect   the   retrogradation    of  the 
moon's  nodes,  and  represent  graphically  the  moon's  orbit  by  a 
method  identical  with  the  method  employed  in  representing  the 
earth's  orbit  (Art.  98),  we  shall  find  the  orbit  to  be  an  ellipse, 
with  the  earth  at  one  of  the  foci.     The  eccentricity  of  the  ellipse 
is  0.0549,  or  very  nearly  -j^th. 

132.  Line  of  Apsides. — That  point  in  the  moon's  orbit  where 
it  is  the  nearest  to  the  earth  is  called  the  perigee,  and  that  point 
where  it  is  the  farthest  from  the  earth,  the  apogee.     The  line 
connecting  these  two  points  is  called  the  line  of  apsides.     It  is 
also  the  major  axis  of  the  moon's  orbit.     This  line  revolves  in  the 
plane  of  the  moon's  orbit  from  west  to  east,  making  a  complete 
revolution  in  very  nearly  nine  years. 

The  following  description  of  the  moon's  orbit,  and  of  the 
changes  to  which  it  is  subject,  is  given  by  Herschel  in  his  Out- 
lines of  Astronomy.  "  The  best  way  to  form  a  distinct  conception 
of  the  moon's  motion  is  to  regard  it  as  describing  an  ellipse  about 
the  earth  in  the  focus,  and  at  the  same  time  to  regard  this  ellipse 
itself  to  be  in  a  twofold  state  of  revolution;  1st,  in  its  own 
plane,  by  a  continual  advance  of  its  own  axis  in  that  plane; 
and  2dly,  by  a  continual  tilting  motion  of  the  plane  itself, 
exactly  similar  to,  but  much  more  rapid  than,  that  of  the  earth's 
equator." 

133.  Variation  in  the  Moons  Meridian  Zenith  Distance. — From 
the  formula  in  Art.  76  we  have, 

z--=L  —  d 
At  any  place,  then,  the  latitude  remaining  constant,  the  least 


DISTANCE    OF   THE    MOON. 


123 


meridian  zenilh  distance  will  occur  when  the  moon's  declination 
has  the  same  name  as  the  latitude,  and  is  at  its  maximum ;  and 
the  greatest  will  occur  when  the  declination  has  the  opposite 
name,  and  is  also  at  its  maximum.  Since  the  plane  of  the 
moon's  orbit  is  inclined,  at  the  most,  5°  20'  to  the  plane  of  the 
ecliptic  (Art.  130),  the  greatest  value  of  the  declination,  either 
north  or  south,  is  5°  20'  -f-  23°  27'.  The  variation  in  the  meri- 
dian altitude  will  therefore  be  double  this  amount,  or  57°  34'.  At 
Annapolis,  in  latitude  38°  59'  N.,  the  greatest  altitude  is  79°  48', 
the  least  22°  14'.  There  is  an  exception  to  this  general  rule  in 
the  case  of  those  places  whose  latitude  is  less  than  28°  47':  since 
at  those  places  the  greatest  altitude  occurs  when  the  moon  is  in 
the  zenith,  or,  as  is  evident  from  the  formula,  when  the  declina- 
tion is  equal  to  the  latitude,  and  has  the  same  name. 

The  new  moon  is  in  the  same  part  of  the  heavens  that  the  sun 
is  in  (Art.  139),  and  the  full  moon  is  in  the  opposite  part.  Since 
the  sun  attains  its  least  altitude  in  winter  and  its  greatest  in 
summer,  new  moons  will  run  low  in  winter  and  full  moons  will 
run  high :  while  in  summer  the  opposite  of  this  will  take  place. 

DISTANCE,   SIZE,    AND   MASS   OF   THE   MOON. 

1 34.  Since  the  sine  of  the  moon's  horizontal  parallax  is  the 


ratio  of  the  radius  of  the  earth  to  the  distance  of  the  nioon  from 
the  earth,  it  is  evident  that  we  can  determine  this  distance  as 
soon  as  we  obtain  the  horizontal  parallax.  The  horizontal 
parallax  may  be  found  in  the  following  manner: — In  Fig.  52,  let 
0  be  the  centre  of  the  earth,  EQ  its  equator,  and  A  and  B  the 
positions  of  two  observers  on  the  same  meridian,  whose  zeniths 


124  DISTANCE   OF   THE    MOON. 

are  Z  and  Z '.  Let  M  be  the  mood's  position  when  crossing  the 
meridian.  The  apparent  zenith  distance  at  A,  corrected  for 
refraction,  is  the  angle  ZAM,  the  geocentric  zenith  distance  is 
ZOM,  and  the  difference  of  these  two  angles,  the  angle  A  MO, 
is  the  parallax  in  altitude.  In  the  same  way  OMB  is  the  pa- 
rallax at  B.  Represent  the  parallax  at  A  by  p,  that  at  B  by 
p,  the  horizontal  parallax  by  P,  the  apparent  meridian  zenith 
distance  at  A  by  z,  and  that  at  B  by  z'. 
We  have,  by  Geometry, 

p  =  z  —  A  0  M, 
p'  =  z'  —  BOM, 
and,  consequently, 

"  p  +  pf  =  z  +  z'  —  AOB. 
We  have  also,  from  Art.  54, 

p  =  P  sin  z, 
p'  =  P  sin  z', 
and,  therefore, 

p  -f-  p'  =  P  (sin  z  -f  sin  z'). 

Combining  the  two  equations,  and  finding  the  expression  for  P, 
_  g  +  z'  —  A  OB 

sin  z  -j-  sin  z' 

But  A  OB  is  evidently  the  difference  of  latitude  of  A  and  B. 
We  have,  then,  as  our  method  of  finding  the  moon's  horizontal 
parallax,  to  subtract  the  difference  of  latitude  of  the  two  places 
from  the  sum  of  the  apparent  zenith  distances,  and  to  divide 
the  remainder  by  the  sum  of  the  sines  of  the  two  zenith  dis- 
tances. 

It  is  important  that  the  two  places  of  observation  shall 
differ  widely  in  latitude.  It  is  not,  however,  necessary  that 
they  shall  be  on  the  same  meridian,  since,  either  from  tables 
of  the  moon's  motion,  or  from  actual  observation  on  successive 
days  before  and  after  the  time  of  observation,  we  can  obtain 
the  change  of  meridian  zenith  distance  corresponding  to  any 
known  difference  of  longitude,  and  thus  reduce  the  two  observed 
zenith  distances  to  the  same  meridian. 

135.  The  Moon's  Horizontal  Parallax. — By  observations  simi- 
lar to  those  above  described,  the  mean  value  of  the  moon's 
equatorial  horizontal  parallax  is  found  to  be  57'  3".  The  mean 


MAGNITUDE   AND    MASS.  125 

distance  of  the  moon  from  the  earth  is  therefore  3962.8  miles 
multiplied  by  the  cosecant  of  57'  3",  or  238,800  miles.  The  hori- 
zontal parallax  varies  between  the  limits  of  61'  32"  and  52'  50", 
and  the  distance  between  257,900  and  221,400  miles.  It  must 
be  noticed  that  by  the  mean  value  of  the  horizontal  parallax 
given  above  is  not  meant  the  half  sum  of  the  two  extreme 
values,  but  the  value  which  the  parallax  has  when  the  moon  is 
at  its  mean  distance  from  the  earth. 

136.  Magnitude  of  the  Moon. — The  angular  semi-diameter  of 
the  moon  at  its  mean  distance  from  the  earth  is  found  to  be 
15'   33."5.     Its  linear  semi-diameter  is  therefore  obtained  by 
multiplying  the  mean  distance  by  the  sine  of   15'  33". 5,  and 
is  found  to  be  1080.8  miles,  or  about  T3Tths  of  the  radius  of  the 
earth.     The  volumes  of  two  spheres  being  to  each  other  as  the 
cubes  of  their  radii,  the  volume  of  the  moon  will  be  found  to 
be  about  ^th  of  that  of  the  earth. 

137.  Mass  of  the  Moon. — The  mass  of  the  moon  is  obtained 
by  the  following  considerations.     If  the  sun  does  not  affect  the 
gravity  of  the  moon  to  the  earth,  and  the  mass  of  the  moon  is 
inappreciable  in  comparison  with  that  of  the  earth,  then  the 
centrifugal  force  of  the  moon  in  its  orbit  ought  exactly  to  equal 
the  attraction  of  the  earth  on  the  moon.     But  if  the  moon  has 
a  sensible  mass,  it  will,  by  the  law  of  gravitation,  attract  the 
earth,  and  its  centrifugal  force  must  be  sufficient  to  counter- 
balance the  sum  of  the  mutual  attractions  of  the  earth  and 
the  moon.     Now,  if  we  refer  to  what  was  demonstrated  in  Art. 
113,  we  find  that  the  moon's  actual  centrifugal  force  is  greater, 
by  -g^th,  than  the  attraction  of  the  earth  at  the  distance  of  the 
moon.     This  would  make  the  mass  of  the  moon  -^gth  of  that  of 
the  earth.     But  it  is  found  that  the  sun  diminishes  sensibly  the 
gravity  of  the  moon  to  the  earth.     This,  therefore,  must  also  be 
taken  into  account,  and  the  resulting  value  of  the  mass  of  the 
moon  is  found  to  be  about  -g-'jSt  of  that  of  the  earth. 

The  density  of  the  moon,  being  directly  as  the  mass,  and 
inversely  as  the  volume,  will  be  |f,  or  about  |ths  of  the  density 
of  the  earth. 

138.  Augmentation  of  the  Moon's  Semi-Diameter. — If  at  any 
lime  we  measure  the  angular  semi-diameter  of  the  moon,  we 


126  PHASES. 

shall  find  that  it  increases  with  the  moon's  altitude,  being  least 
when  the  moon  is  in  the  horizon  and  greatest  when  in  the 
zenith.  This  increase  is  explained  in  Fig.  53.  Let  E  be  the 

centre  of  the  earth,  and 
M that  of  the  moon.  With 
the  distance  between  j£and 
M  as  a  radius,  describe  the 
semi-circumference  AM' B. 
When  the  moon  is  in  the 
horizon  of  the  point  C,  its 
distances  from  C  and  from 
E  are  very  nearly  equal. 
53.  But  as  the  moon  rises,  the 

distance  CM  continually  decreases,  while  EM,  the  distance  of  the 
moon  from  the  earth's  centre,  remains  sensibly  constant.  When 
the  moon  is  in  the  zenith,  or  at  M',  the  distance  CM'  is  less  than 
EM'  by  the  radius  of  the  earth.  Now,  the  angular  semi-diame- 
ter of  the  moon  will  increase,  as  shown  in  Art.  98,  very  nearly 
as  the  distance  of  the  moon  from  the  observer  decreases.  But 
the  earth's  radius  is  about  ^Oth  of  the  distance  of  the  moon 
from  the  earth's  centre:  therefore  the  semi-diameter  of  the 
moon  in  the  zenith  will  be  greater  than  the  semi  diameter  in 
the  horizon  by  -g-^th  of  itself,  or  by  about  15".  This  increase 
is  called  the  augmentation  of  the  moon's  semi-diameter. 

THE   MOON'S   PHASES. 

139.  Two  bodies  are  said  to  be  in  conjunction  when  they  have 
the  same  longitude.  They  are  said  to  be  in  opposition  when 
their  longitudes  differ  by  180° ;  and  in  quadrature  when  their 
longitudes  differ  by  either  90°  or  270°. 

The  moon  is  an  opaque  body,  which  is  rendered  visible  to 
us  by  the  rays  of  light  which  it  reflects  from  the  sun.  The 
phases  of  the  moon  are  due  to  the  different  relative  positions 
to  the  sun  and  the  earth  which  it  has  while  revolving  about 
the  earth. 

In  Fig.  54  let  #be  the  earth,  and  the  circle  ACFH  the  orbit 
of  the  moon.  Since  the  inclination  of  the  plane  of  the  moon's 
orbit  to  the  plane  of  the  ecliptic  is  only  a  few  degrees,  we  may 


PHASES. 


127 


neglect  it  in  this  case,  and  suppose  the  two  planes  to  coincide. 
Let  the  sun  lie  in  the  direction  ES.     Since  the  distance  of  the 


Fig.  54. 

sun  from  the  earth  is  about  387  times  the  distance  of  the  moon 
from  the  earth,  the  lines  JES,  ff/S,  J3S,  &c.,  drawn  to  the  sun 
from  different  points  of  the  moon's  orbit,  may  be  considered  to  be 
sensibly  parallel.  Let  us  first  suppose  the  moon  to  be  in  con- 
junction with  the  sun  at  the  point  A.  Here  only  the  dark 
portion  ot  the  moon  is  turned  towards  the  earth,  and  the  moon 
is  therefore  invisible.  This  is  called  new  moon.  As  the  moon 
moves  on  towards  B,  the  enlightened  part  begins  to  be  visible, 
and  when  it  reaches  C,  90°  in  longitude  from  the  sun,  half  the 
enlightened  part  is  visible,  and  the  moon  is  at  its  first  quarter. 
When  the  moon  is  at  F,  in  opposition  to  the  sun,  all  the  illumi- 
nated part  is  turned  towards  the  earth,  and  the  moon  is  full. 
The  moon  wanes  after  leaving  F,  passes  through  its  laat  quarter 
at  ff,  and  finally  becomes  again  invisible. 

Between  A  and  C  the  moon  is  crescent,  as  represented  at  L, 
and  between  C  and  F  it  is  gibbous,  as  represented  at  N.  The 
same  terms  are  also  applied  to  the  appearance  of  the  moon  be- 
tween H  and  A  and  between  F  and  H. 

140.  Phases  of  the  Earth  to  the  Moon. — It  is  evident  from  Fig. 
54  that  the  earth  presents  phases  to  the  moon  identical  in  cha- 
racter with  those  presented  by  the  moon  to  the  earth,  although 


128 


SIDEREAL   AND   SYNODICAL   PERIODS. 


similar  phases  are  Dot  presented  by  each  body  at  the  same  time. 
Thus  at  the  time  of  new  moon  tlie  earth  is  full  to  the  moon : 
and  the  light  which  it  then  reflects  to  the  moon  renders  the 
unenlightened  part  of  the  moon  faintly  visible  to  the  earth.  As 
the  moon  moves  on  to  its  first  quarter,  the  earth  reflects  less  and 
less  light  to  it,  until  finally  the  unenlightened  portion  disappears. 

SIDEREAL   AND   SYNODICAL   PERIODS. 

141.  The  sidereal  period  of  the  moon  is  the  interval  of  time 
in  which  it  makes  one  complete  revolution  in  its  orbit  about 
the  earth.  The  synodieal  period  (or  lunation)  is  the  interval 
between  two  successive  conjunctions  or  oppositions.  Owing  to 
the  earth's  revolving  about  the  sun,  and  carrying  the  moon  with 
it,  the  synodieal  period  is  longer  than  the  sidereal  period,  as  may 
be  seen  in  Fig.  55. 

Let  S  be  the  sun,  E  the 
earth,  and  MANB  the  or- 
bit of  the  moon.  Let  the 
moon  be  at  M,  in  conjunc- 
tion with  the  sun.  As  the 
moon  moves  about  E  in 
the  curve  MANB,  the 
earth  also  moves  about  the 
sun  in  the  direction  EE'. 
The  next  conjunction  will 
therefore  not  occur  until  the 
moo;n  reaches  M".  Now, 
if  through  E'  we  draw  the 
line  M'N'  parallel  to  MNt  the  sidereal  period  of  the  moon  is 
completed  when  the  moon  reaches  M'.  The  synodieal  period  is 
therefore  greater  than  the  sidereal  period  by  the  time  required 
by  the  moon  to  pass  through  the  angle  M'E'M".  This  angle  is 
evidently  equal  to  the  angle  ESE',  which  is  the  angular  advance 
of  the  earth  in  its  orbit  in  the  period  of  one  synodieal  revo- 
lution of  the  moon.  In  one  lunar  month,  then,  the  angular  ad- 
vance of  the  moon  in  its  orbit  is  greater  by  360°  than  the  angu- 
lar advance  of  the  earth  in  its  orbit.  If,  therefore,  we  denote 
the  moon's  sidereal  period  in  days  by  P,  its  synodieal  period  by 


Fig.  55. 


SIDEREAL   AND   SYNODICAL    PERIODS.  129 

S,  and  the  earth's  sidereal  period,  or  one  sidereal  year,  by  T,  we 
shall  have, 

360° 

— — -  =  the  earth's  daily  angular  velocity, 

360° 

— p—  =  the  moon's  daily  angular  velocity, 

360° 

— — -  =  the  moon's  daily  angular  gain  on  the  earth, 

Hence  we  shall  have, 

360°       360°       360°. 


> 


P  T  S 

ST 

~  S+T 

The  sidereal  period  of  the  moon  is  therefore  obtained  by  mul- 
tiplying the  sidereal  year  by  the  moon's  synodical  period,  and 
dividing  the  product  by  the  sum  of  the  sidereal  year  and  the 
synodical  period. 

142.  Values  of  the  Synodical  and  Sidereal  Periods. — The  value 
of  the  synodical  period  is  not  constant,  but  varies  from  month 
to  month.     A  mean  value  may,  however,  be  obtained  by  divid- 
ing the  interval  of  time  between    two  oppositions,  not  conse- 
cutive, by  the  number  of  synodical  revolutions  in  that  interval. 
Now,  the  day,  the  hour,  and  even  the  probable  minute,  at  which 
an  opposition  of  the  moon  occurred  in  the  year  720  B.C.,  were 
recorded  by  the  Chaldseans;  and  by  comparing  this  time  with 
the  results  of  recent  observations,  an  extremely  accurate  value 
of  the  mean  synodicaT  period  is  obtained      It  is  found  to  be 
29d.  12h.  44m.  3s.     We  have,  then,  for  the  value  of  the  side- 
real period,  by  the  formula  in  the  preceding  article, 

_  365.256  X  29.53 
~  3657256  -r-  29.53      JS 
whence  we  obtain  the  value  already  given  in  Art.  128. 

143.  Retardation  of  the  Moon,  and  the  Harvest  Moon. — The 
mean  daily  motion  of  the  moon  towards  the  east  is  about  13°, 
while  that  of  the  sun  is,  as  we  have  already  seen,  about  1°: 
hence  the  moon  is  continually  falling  to  the  rear  of  the  sun  in 
apparent  westward  motion,  and  the  interval  of  time  between  any 
two  successive  transits  of  the  moon  is  greater  than  the  similar 


1<50  ROTATION    OF    THE    MOON. 

interval  in  the  case  of  the  sun.  r£he  moon,  therefore,  rises  later, 
and  sets  later,  day  by  day.  This  is  called  the  retardation  of  the 
moon.  Its  amount  varies  considerably  in  value,  but  is  on  the 
average  about  fifty  minutes. 

The  less  the  angle  which  the  plane  of  the  moon's  orbit  makes 
with  the  plane  of  the  horizon,  the  less  does  the  advance  of  the 
moon  carry  it  with  reference  to  the  horizon,  and,  consequently, 
the  less  is  the  retardation  of  the  moon  in  rising.  Now,  since 
the  moon's  orbit  very  nearly  coincides  with  the  ecliptic,  the 
retardation  in  rising  will  in  general  be  the  least,  when  the 
ecliptic  makes  the  least  angle  with  the  horizon.  By  reference 
to  a  celestial  globe,  it  will  be  seen  that  the  ecliptic  makes  the 
least  angle  with  the  horizon  when  the  vernal  equinox  is  in  the 
eastern  horizon.  The  least  retardation  in  rising,  therefore, 
occurs  in  each  month  when  the  moon  is  near  the  sign  of  Aries. 
This  least  retardation  is  especially  noticeable  when  it  occurs  at 
the  time  of  full  moon.  Now,  when  the  moon  is  in  Aries,  and 
full,  the  sun  must  be  in  Libra,  or  near  the  autumnal  equinox. 
This  occurs  about  the  21st  of  September.  About  the  time,  then, 
of  the  full  moon  which  occurs  near  the  21st  of  September,  the 
moon  will  rise,  for  two  or  three  nights,  only  about  half  an  hour 
later  each  night.  Usually  this  small  retardation  is  noticed  at 
the  times  of  two  full  moons,  one  in  September  and  the  other  in 
October.  The  first  is  called  the  Harvest  Moon,  the  second  the 
Hunter's  Moon.  All  this  relates  to  the  Northern  Hemisphere. 

ROTATION.    LIBRATIONS     AND    OTHER   PERTURBATIONS. 

144.  Rotation  of  the  Moon. — By  observation  of  the  spots  upon 
the  disc  of  the  moon,  it  is  found  that  very  nearly  the  same 
surface  of  the  moon  is  turned  continually  towards  the  earth. 
The  conclusion  drawn  from  this  fact  is  that  the  moon  rotates 
upon  an  axis  in  the  same  time  in  which  it  revolves  about  the 
earth,  or  in  27.3  days.  The  plane  in  which  this  rotation  is 
performed  makes  an  angle  of  about  1°  32'  with  the  plane  of  the 
ecliptic. 

If  there  are  any  inhabitants  of  the  moon,  their  day  will  be 
equal  in  length  to  about  twenty-nine  of  our  days,  and  their 
ni^ht  to  about  twenty-nine  of  our  nights.  Since  the  plane  of 


LIBRATIONS    OF    THE    MOON. 


131 


the  moon's  equator  is  so  nearly  coincident  with  the  plane  of  tho 
ecliptic,  there  will  hardly  be  any  sensible  change  of  seasons : 
or  if  there  is,  the  lunar  day  will  be  the  lunar  summer,  and  the 
night  the  winter.  To  the  inhabitants  of  one  hemisphere  the 
earth  will  be  perpetually  invisible,  while  to  the  inhabitants  of 
the  other  hemisphere  it  will  present  the  appearance  of  a. body 
very  nearly  stationary  in  their  sky,  exhibiting  phases  similar 
to  those  which  we  see  in  the  moon,  with  a  radius  nearly  four 
times  that  of  the  moon,  and  a  surface  about  thirteen  times  that 
of  the  moon. 

145.  Librations. — By  libration  is  meant  an  apparent  oscilla- 
tory movement  of  the  moon,  which  enables  us,  in  the  course  of 
its  revolution,  to  see  something  more  than  an  exact  hemisphere. 

The  libration  in  longitude  is  due  to  the  fact  that  the  moon's 
rotation  on  its  axis  is  perfectly  uniform,  while  its  motion  about 
the  earth  is  not.  Hence  the  line  drawn  from  the  centre  of  the 


Fig.  56. 


earth  to  that  of  the  moon  does  not  always  intersect  the  surface 
of  the  moon  at  the  same  point,  and  we  are  able  at  times  to  look 


132  OTHER    PERTURBATIONS. 

a  few  degrees,  east  or  west,  beyond  the  mean  visible  border.  If, 
in  Fig.  56,  ABCD  represents  the  earth,  E  its  centre,  and  R 
the  centre  of  the  rnoon,  the  dotted  lines  at  iV  denote  the  limits 
between  which,  as  the  moon  revolves  about  the  earth,  the  visible 
border  may  deviate  from  its  mean  position. 

The  libration  in  latitude  is  due  to  the  fact  that  the  axis  of  the 
moon,  remaining  constantly  parallel  to  itself,  is  not  perpen- 
dicular to  the  plane  of  the  moon's  orbit,  but  is  inclined  to  it  at  an 
angle  of  about  83°  19'.  We  are  therefore  able  at  certain  times 
to  see  about  6°  41'  beyond  the  north  pole  of  the  moon,  and  at 
other  times  the  same  amount  beyond  the  south  pole.  Thus  in 
Fig  56,  when  the  moon  is  at  M,  we  can  see  beyond  the  pole  P, 
and  when  the  moon  is  at  0,  beyond  the  pole  p:  since  in  each 
case  we  can  see  nearly  that  portion  of  the  moon  which  lies  be- 
tween the  earth  and  the  circle  ab,  whose  plane  is  perpendicular 
to  the  plane  of  the  moon's  orbit. 

The  diurnal  libration  is  due  to  the  difference  between  that 
hemisphere  of  the  moon  which  is  turned  towards  the  centre  of 
the  earth  and  that  which  is  turned  towards  any  point  on  the 
surface.  When,  for  instance,  the  moon  is  at  Z/,  an  observer  at 
C  will  see  the  same  hemisphere  which  is  turned  towards  the 
earth's  centre,  while  an  observer  at  G  will  see  a  different  one. 
The  hemisphere  which  is  turned  towards  any  observer  when 
the  moon  is  rising  will  also  be  different  from  the  one  which  is 
turned  towards  him  when  the  moon  is  setting.  It  is  evident  in 
the  figure  that  the  amount  of  this  libration  varies  with  the  angle 
ERG;  that  is  to  say,  with  the  moon's  parallax. 

Notwithstanding  all  these  librations,  we  are  able  to  see  in  all 
only  about  fV0ths  of  the  moon's  surface,  according  to  Arago: 
the  remainder  being  continually  concealed  from  our  view. 

146.  Other  Perturbations. — Besides  these  librations,  and  the 
perturbations  already  mentioned  (Art.  132),  there  are  other  per- 
turbations in  the  moon's  longitude  of  which  only  a  very  brief  notice 
can  here  be  given.  The  greatest  of  these  perturbations  is  called 
evection,  and  was  discovered  by  Ptolemy  in  the  second  century. 
It  arises  from  the  variation  in  the  eccentricity  of  the  moon's 
orbit,  and  from  the  fluctuations  in  the  general  advance  of  the 
line  of  the  apsides.  By  it  the  moon's  mean  longitude  is  alternately 


LUNAR   CYCLE.  133 

increased  and  decreased  by  about  1°  20'.  Another  perturbation 
in  the  moon's  motion  is  called  variation.  It  depends  solely  on 
the  angular  distance  of  the  moon  from  the  sun,  and  its  maximum 
is  37'.  The  annual  equation  depends  on  the  variable  distance 
of  the  earth  from  the  s'in,  and  amounts  to  11'.  The  secular  acce- 
leration is  an  increase  in  the  moon's  motion  which  has  been  going 
on  for  many  centuries,  at  the  rate  of  about  10"  a  century.  This 
perturbation  is  partly  due  to  the  diminution  of  the  eccentricity 
of  the  earth's  orbit;  and  from  what  has  been  said  on  that  subject 
in  Art.  98,  it  is  evident  that  this  inequality  will  at  some  distant 
day  become  a  secular  retardation.  [See  Note,  page  154.] 

All  of  these  perturbations  are  satisfactorily  explained  by  the 
investigation  of  what  is  known  as  the  problem  of  the  three  bodies, 
in  which  two  bodies  are  supposed  to  revolve  about  their  common 
centre  of  gravity,  according  to  the  law  of  universal  gravitation, 
and  the  effects  of  the  attraction  exerted  by  a  third  body  upon 
the  motions  of  these  two  bodies  are  made  the  object  of  mathe- 
matical examination. 

THE    LUNAR   CYCLE. 

147.  If  we  multiply  the  number  of  days,  hours,  &c.,  in  a 
synodical  period  of  the  moon  (Art.  142)  by  235,  the  product 
will  be  6939d.  16h.  27m.  50s.  Now,  in  a  period  of  nineteen  civil 
years  there  are  either  6939  days,  or  6940  days,  according  as  there 
are  four  or  five  leap  years  in  that  period.  If,  then,  in  any  year, 
new  moon  occurs  on  any  particular  day  of  the  month,  the  first 
of  January,  for  instance,  it  will  occur  again  on  the  first  of  Janu- 
ary (or  at  all  events  within  a  few  hours  of  its  end  or  beginning), 
after  an  interval  of  nineteen  years ;  and  all  the  new  moons  and 
the  other  phases  will  occur  on  very  nearly  the  same  days  through- 
out the  second  period  of  nineteen  years  on  which  they  occurred 
during  the  first  period.  This  period  is  called  the  Lunar  Cycle. 
It  is  also  called  the  Metonic  Cycle,  having  been  originally  dis- 
covered, B.C.  432,  by  Meton,  an  Athenian  mathematician.  The 
present  lunar  cycle  began  in  1862. 

This  cycle  is  used  in  finding  Easter:  Easter  being  the  first 
Sunday  after  the  full  moon  which  occurs  either  upon  or  next 
after  the  21st  day  of  March. 


134  APPEARANCE   OF    THE    MOON. 

The  golden  number  of  any  ye%r  is  the  number  which  marks 
the  place  of  that  year  in  the  cycle.  It  may  be  found  for  any 
year  by  adding  1  to  the  number  of  that  year,  and  dividing  the 
sum  by  19 ;  the  remainder  (or  19,  if  there  is  no  remainder)  is 
the  golden  number. 

Four  lunar  cycles,  or  seventy-six  civil  years,  constitute  what 
is  called  the  Callippic  cycle. 

GENERAL   DESCRIPTION   OF   THE   MOON. 

148.  When  viewed  through  powerful  telescopes,  the  surface 
of  the  moon  is  found  to  be  made  up  of  mountains,  valleys,  and 
plains,  similar  in  general  appearance  to  those  that  exist  on  the 
earth.  As  a  whole,  however,  the  surface  of  the  moon  is  much 
more  uneven  than  that  of  the  earth.  The  heights  of  over  1000 
lunar  mountains  have  been  measured,  and  some  of  them  have 
been  found  to  exceed  20,000  feet.  Many  of  these  mountains 
bear  the  appearance  of  having  been  at  one  time  volcanoes,  far 
surpassing  in  size  and  activity  those  on  the  earth.  The  common 
belief  among  astronomers  seems  to  be  that  these  lunar  volcanoes 
are  now  extinct.  Messrs.  Beer  and  Madler,  two  Prussian  astron- 
omers who  have  made  the  moon  their  special  study,  have  de- 
tected no  signs  of  activity  in  any  of  the  volcanoes  which  they  have 
examined.  A  few  years  ago,  certain  phenomena  were  noticed 
which  seemed  to  show  that  one  at  least  of  these  volcanoes, 
named  Linne",  is  not  extinct :  but  later  observations  do  not  confirm 
this  suspicion. 

There  are  no  signs  of  the  existence  of  water  on  the  moon. 
Certain  large  dark  patches  are  seen,  which  were  formerly  con- 
sidered to  be  oceans,  gulfs,  &c.,  and  were  so  named ;  but  increased 
telescopic  power  shows  that  they  are  dry  plains. 

It  seems  to  be  still  an  open  question  whether  or  not  the  moon 
has  an  atmosphere.  If  there  is  an  atmosphere,  it  must  be  of  an 
extremely  minute  height  and  density ;  for  we  see  no  clouds  and 
no  twilight,  and  there  is  nothing  in  the  phenomena  of  the  occul- 
tations  of  stars  by  the  moon  which  shows  the  existence  of  even 
the  rarest  atmosphere.  Some  observers,  however,  and  among 
them  Messrs.  Beer  and  Madler,  believe  that  they  have  detected 
signs  of  the  existence  of  a  very  slight  atmosphere. 


PLATE  II 


TOTAL  ECLIPSE  OF  THE  SUN,  OF  JULY  18, 1860, 
showing  the  Corona  and  the  Red  Flames;    as  observed  by  Dr. 
Feilitzsch,  at  Castellon  de  la  Plana 


LUNAR   ECLIPSE.  135 


CHAPTER  X. 

LUNAR   AND   SOLAR   ECLIPSES.      OCCULTATIONS. 

149.  Eclipses. — The  obscuration,  either  partial  or  total,  of  the 
light  of  one  celestial  body  by  another  is  in  astronomy  termed  an 
eclipse.  When  the  earth  comes  between  the  sun  and  the  moon, 
the  light  of  the  sun  is  shut  off  from  the  moon,  and  we  have  a 
Mjnar  eclipse.  A  lunar  eclipse  can  occur  only  at  the  time  when 
the  moon  is  in  opposition  to  the  sun,  that  is  to  say,  at  the  time 
:>f  full  moon.  When  the  moon  comes  between  the  earth  and  the 
sun,  the  light  of  the  sun  is  shut  off  from  the  earth,  and  we  have 
a  solar  eclipse.  A  solar  eclipse  can  occur  only  at  the  time  of 
new  moon.  An  eclipse  of  a  star  or  a  planet  by  the  moon  is  called 
an  occultation. 

If  the  orbit  of  the  moon  lay  in  the  plane  of  the  ecliptic,  a 
lunar  and  a  solar  eclipse  would  occur  in  every  month.  Owing, 
however,  to  the  inclination  of  the  plane  of  the  moon's  orbit  to 
the  plane  of  the  ecliptic,  the  latitude  of  the  moon  is  usually  too 
great  to  allow  either  kind  of  eclipse  to  take  place;  and  it  is  only 
in  special  cases,  when  the  moon  is  in  or  near  the  plane  of  the 
ecliptic  at  the  time  of  conjunction  or  opposition,  that  an  eclipse 
of  the  sun  or  the  moon  is  possible. 


LUNAR   ECLIPSE. 

150.  In  Fig.  57  let  S  be  the  centre  of  the  sun,  and  E  that  of 


Fig.  57. 


136  LUNAR    ECLIPSE. 

the  earth.  Draw  the  lines  BH.  and  GG,  tangent  to  the  two 
spheres.  These  lines  will  meet  at  some  point  A,  and  AHEG 
will  be  a  section  of  the  shadow  cast  by  the  earth. 

The  whole  shadow  is  of  a  conical  shape,  the  vertex  of  the  cone 
being  at  A ;  and  a  lunar  eclipse  will  occur  whenever  the  moon  is 
within  this  shadow.  Draw  the  tangent  lines  BG  and  CH. 
KDL  is  a  section  of  a  second  cone  whose  vertex  is  at  Z>.  The 
earth's  shadow  is  called  the  umbra,  and  that  portion  of  the  second 
cone  which  lies  outside  of  the  umbra  is  called  the  penumbra. 
Thus  KHA  and  A  GL  are  sections  of  the  penumbra.  It  must 
be  noticed,  in  regard  to  the  construction  of  this  figure,  that  since 
only  one  tangent  can  be  drawn  to  the  circumference  of  a  circle 
at  any  one  point,  the  lines  BG  and  CH  do  not  touch  the  two 
circumferences  at  precisely  the  same  points  at  which  BH  and 
CG  touch ;  and  that,  furthermore,  in  all  these  figures  the  relative 
size  of  the  sun  should  be  immensely  greater  than  it  is. 

Now  let  M'MM"  represent  a  portion  of  the  moon's  orbit  at  the 
time  of  a  lunar  eclipse.  As  soon  as  the  moon  passes  within  the 
line  DK,  some  of  the  rays  of  the  sun  will  be  cut  off  from  it  by 
the  earth,  and  its  brightness  will  begin  to  decrease.  The  whole 
disc,  however,  will  still  be  visible.  As  soon  as  the  moon  begins 
to  pass  within  the  line  HA,  the  disc  will  begin  to  disappear,  and 
when  the  whole  disc  has  passed  within  the  cone,  the  eclipse  will 
be  total. 

151.  Different  Kinds  of  Eclipses. — When    the  moon's  orbit 
is  so  situated  that  only  a  part  of  the  moon  enters  the  umbra, 
we  have  a  partial  eclipse.     When  the  moon  does  not  enter  the 
umbra,  but  merely  touches  it,  we  have  an  appulse.     When  the 
centre  of  the  moon  coincides  with  the  line  which  connects  the 
centre  of  the  earth  and   that  of  the  sun,  the  eclipse  is   cen- 
tral.    A  central  eclipse  occurs  very  rarely,  if  indeed  it  occurs 
at  all. 

152.  The  Semi-Angle  of  the  Umbral  Cone. — The  semi-angle  of 
the  umbral  cone  is  the  angle  EA  G,  Fig.  58.     Now  we  have,  by 
Geometry, 

SEC=  ECG  +  EAG. 
But  SEC  is  the  sun's   angular  semi-diameter,  and  EGG  is  its 


LUNAR   ECLIPSE.  137 

horizontal  parallax.     Putting  S  for  SEC,  and  P  for  ECG,  we 
have, 

EAG  =  S— P. 


Fig.  58. 

153.  The  Angular  Semi-Diameter  of  the  Shadow  at  the  Distance 
of  the  Moon. — The  angular  semi-diameter  of  the  shadow  at  the 
distance  of  the  moon  is  the  angle  MEM'.   We  have,  by  Geometry, 

EM'  G  =  MEM'  +  EA  G. 

Now  EM'  G  is  the  moon's  horizontal  parallax,  which  we  will 
represent  by  P',  and  the  value  of  EA  G  has  been  obtained  in 
the  preceding  article.  We  therefore  have, 

MEM'  =  P'+P  — S. 

Observation  shows  that  the  earth's  atmosphere  increases  the 
apparent  breadth  of  the  shadow  by  about  its  one-fiftieth  part : 
hence  in  practice  the  angular  semi-diameter  of  the  shadow  is 
taken  equal  to  f  J  (P'  -f-  P  —  S).  If  we  substitute  in  this  expres- 
sion the  least  values  of  Pf  and  P,  and  the  greatest  value  of  JSt 
from  the  table  given  in  Art.  155,  we  shall  find  that  the  least 
value  of  the  angular  semi-diameter  of  the  shadow  is  about  37'  25" : 
so  that  the  entire  breadth  of  the  shadow  is  always  more  than 
double  the  greatest  diameter  of  the  moon. 

154.  Length  of  the  Earth's  Shadow. — The  length  of  the  shadow, 
or  the  line  EA,  can  be  computed  from  the  right-angled  triangle 
EA  G,  in  which  we  have, 

EA=JEQco8ec(8  —  P). 

The  mean  value  of  this  length  is  858,000  miles,  or  more  than 
three  times  the  distance  of  the  moon  from  the  earth. 

155.  Lunar  Ecliptic  Limits. — We  see  from  Fig.  58  that  a  lunar 
eclipse  can  occur  only  when  the  moon's  geocentric  latitude  at 
the  time  of  opposition,    (or  at  full  moon,)  is  less  than  the  sum 
of  the  angular  semi-diameter  of  the  shadow  and  the  semi-diameter 


138 


LUNAR   ECLIPSE. 


of  the  moon.     If  we  represent  the.  moon's  semi-diameter  by 
the  expression  for  this  sum  is 


If  the  moon's  geocentric  latitude  at  the  time  of  opposition  is 
greater  than  the  greatest  value  which  this  expression  can  attain, 
no  eclipse  can  possibly  occur :  if  it  is  less  than  the  least  value  of 
the  expression,  an  eclipse  is  inevitable.  These  two  values  of 
this  expression  are  called  the  lunar  ecliptic  limits.  Now,  we  have 
by  observation  the  following  values  of  P,  P'}  &c. : 


MAXIMA. 

MINIMA. 

P' 

61'  32" 

52'  50" 

p 

9 

9 

S' 

16    46 

14    24 

s 

16    18 

15    45 

In  order  to  find  the  greatest  value  of  the  expression,  we  sub- 
stitute in  it  the  greatest  values  of  P,  P  and  $',  and  the  least  value 
of  S.  The  result  is  1°  3'  37":  and  no  eclipse  will  occur  when  the 
moon's  latitude  exceeds  this  limit.  The  least  value  of  the  expres- 
sion is  51'  49":  and  when  the  moon's  latitude  at  opposition  is  less 
than  this,  an  eclipse  cannot  fail  to  occur.  There  are  some  con- 
siderations, however,  which  have  not  been  taken  into  account, 
which  may  increase  each  of  these  limits  by  about  16". 

When  the  moon's  latitude  at  opposition  is  within  these  limits, 
an  eclipse  is  possible,  but  not  necessarily  certain.  In  order  to 
determine  whether  in  such  case  it  will  or  will  not  occur,  the 
actual  values  which  P,  P',  S  and  S'  will  have  at  that  time  must 
be  substituted  in  the  expression,  and  the  result  compared  with 
the  corresponding  latitude  of  the  moon. 

156.  Since  a  lunar  eclipse  is  caused  by  the  moon's  entering 
the  earth's  shadow,  it  will  be  seen  at  the  same  instant  of  time  by 
every  observer  who  has  the  moon  above  his  horizon :  and  the 
character  of  the  eclipse,  whether  total  or  partial,  will  be  every- 


SOLAR    ECLIPSE.  139 

where  the  same,  As  the  moon's  motion  towards  the  east  is  more 
rapid  than  that  of  the  earth  (and  consequently  of  the  shadow), 
the  eclipse  will  begin  at  the  eastern  limb  of  the  moon.  A  total 
eclipse  of  the  moon  may  last  for  nearly  two  hours.  Even  when 
totally  eclipsed,  however,  the  moon  does  not,  in  general,  disappear 
from  view,  but  shines  with  a  dull  reddish  light.  This  phenomenon 
is  caused  by  the  earth's  atmosphere,  which  refracts  the  rays  of 
light  from  the  sun  which  enter  it  near  the  points  G  and  H,  Fig. 
57,  and  turns  them  into  the  cone.  The  rays  which  pass  still 
nearer  to  these  points  are  probably  absorbed  by  the  atmosphere, 
thus  giving  rise  to  the  observed  increase  of  the  shadow  mentioned 
in  Art.  153. 

SOLAR   ECLIPSE. 

157.  In  Fig.  59,  let  S  represent  the  sun,  E  the  earth,  and  M 
the  moon,  at  the  time  of  a  solar  eclipse.  HAK  will  be  a  section 
of  the  moon's  umbra,  and  GHA  and  AKD,  sections  of  its  pen- 
umbra. 


Fig.  59. 

To  an  observer  situated  within  the  umbra,  at  any  point  of  the 
arc  dby  the  eclipse  will  be  total ;  while  to  one  situated  within  the 
penumbra,  as  at  L,  for  instance,  the  eclipse  will  be  partial. 
Beyond  the  penumbra  no  eclipse  whatever  will  be  seen.  Hence 
the  geographical  position  of  the  observer  determines  the  charac- 
ter of  the  eclipse  :  a  condition  different  from  that  in  the  case  of 
a  lunar  eclipse,  which  we  have  seen  is  the  same  to  all  observers. 

158.  Length  of  the  Moon's  Shadow.— It  is  evident  that,  to  an 
observer  at  the  apex  of  the  shadow  A,  the  angular  semi-diametera 
of  the  sun  and  the  moon  would  be  equal.  Now,  the  mean  angular 
semi-diameters  of  these  two  bodies  as  seen  from  the  earth's  centre 


140  SOLAR    ECLIPSE. 

are  nearly  equal ;  hence  the  mean  position  of  the  apex  does  not  fall 

very  far  from  the  earth's  centre  E.    An  approximate  value  of  the 

length  of  the  shadow  may  be  thus  obtained.    We  have  in  Fig.  59, 

.     _.,_     HM       CS 

8mHAM=AM=AS' 

But  we  have  just  now  seen  that  HAM  is  the  sun's  angular 
genii-diameter,  as  seen  from  A ;  and  as  AE  is  small  compared 
with  AS,  we  may  consider  the  angle  HAM  to  be  the  sun's  geo- 
centric semi-diameter.  Denoting  this  by  S,  we  have, 

sin  £  =  — — . 
AM 

Now,  if  Sf  represents  the  moon's  geocentric  semi-diameter,  we 
have, 

HM 


Combining  these  two  equations,  and  finding  the  value  of  AM, 
we  have, 


Knowing,  then,  the  distance  of  the  moon  from  the  earth's  centre, 
and  the  semi-diameters  of  the  sun  and  the  moon,  we  may  find 
the  length  of  AM.  When  the  two  semi-diameters  are  equal,  we 
have  AM  equal  to  EM,  and  the  apex  is  at  the  earth's  centre. 
When  the  semi-diameter  of  the  moon  is  greater  than  that  of 
the  sun,  the  apex  falls  beyond  the  earth's  centre  :  when  it  is 
less,  the  apex  does  not  reach  the  centre.  Appropriate  calcula- 
tions will  show  that  when  both  sun  and  moon  are  at  their  mean 
distances  from  us,  the  apex  falls  short  of  the  earth's  surface  : 
and  that  when  the  moon  is  at  its  least  distance  from  the  earth, 
and  its  shadow  is  the  longest,  the  apex  falls  about  14,000  miles 
beyond  the  earth's  centre. 

159.  Different  Kinds  of  Eclipses.  —  When  the  shadow  falls 
beyond  the  earth's  surface,  the  eclipse  is  total,  as  we  have 
already  seen,  within  the  umbra,  and  partial  within  the  penumbra. 
When  the  apex  just  touches  the  earth,  the  eclipse  is  total  only 
at  the  point  where  it  touches.  When  the  apex  falls  short  of 
the  surface,  there  will  be  no  total  eclipse  ;  but  at  the  point  in 
which  the  axis  of  the  cone,  prolonged,  meets  the  earth,  the 


SOLAR   ECLIPSE.  141 

observer  will  see  what  is  called  an  annular  eclipse,  the  moon 
being  projected  upon  the  disc  of  the  sun,  but  not  covering  it. 

160.  Solar  Ecliptic  Limits. — In  Fig.  60  let  S  represent  the 
sun,  E  the  earth,  and  M  the  moon.  No  eclipse  of  the  sun  can 
occur  unless  some  part  of  the  moon  passes  within  the  lines  BG 
B 


G 

Fig.  60. 

and  GD,  drawn  tangent  to  the  sun  and  the  earth :  that  is,  unless 
the  moon's  geocentric  latitude  is  less  than  the  angle  MES. 
Now  we  have, 

MES  =  MEA  +  AEB  +  BES, 
and,  also, 

AEB=  CAE—  CBE. 

BES  is  the  sun's  semi-diameter,  MEA  that  of  the  moon,  CAE 
the  moon's  horizontal  parallax,  and    CBE  the  sun's :   hence, 
using  the  notation  already  employed  in  Art.  155,  we  have, 
MES  =  8  +  8  +  P.  — P. 

The  greatest  value  of  this  expression  is  found  by  employing 
the  greatest  values  of  S,  $',  and  P',  and  the  least  value  of  P, 
as  given  in  Art.  155,  and  is  1°  34'  27" :  and  there  will  be  no 
eclipse  if  the  moon's  latitude  at  conjunction  is  greater  than 
this  amount.  The  least  value  is  1°  22'  50"  ;  and  if  the  latitude 
at  conjunction  is  less  than  this  an  eclipse  is  inevitable.  These 
two  values,  which,  owing  to  certain  considerations  omitted  in 
this  discussion,  should  both  be  increased  by  about  25",  are  called 
the  solar  ecliptic  limits.  In  order  to  determine  whether  an 
eclipse  will  occur  when  the  moon's  latitude  at  conjunction  falls 
within  these  limits,  we  must  substitute  in  the  expression  the 
values  which  the  different  quantities  will  really  have  at  that 
time,  and  compare  the  result  with  the  corresponding  latitude 
of  the  moon. 

161.    General  Phenomena. — Since   the  moon  moves   towards 


142  CYCLE   OF   ECLIPSES. 

the  east  more  rapidly  than  the  sun,  a  solar  eclipse  will  begin 
at  the  western  side  of  the  sun.  For  the  same  reason  the  moon's 
shadow  will  cross  the  earth  from  west  to  east,  and  the  eclipse 
will  begin  earlier  at  the  western  portions  of  the  earth's  surface 
than  at  the  eastern.  The  moon's  penumbra  is  tangent  to  the 
earth's  surface  at  the  beginning  and  the  end  of  the  eclipse,  so 
that  the  sun  will  be  rising  at  that  place  where  the  eclipse  is 
first  seen,  and  setting  at  the  place  where  it  is  last  seen.  A  solar 
eclipse  may  last  at  the  equator  about  4J  hours,  and  in  these 
latitudes  about  3.\  hours.  That  portion  of  the  eclipse,  however, 
in  which  the  sun  is  wholly  concealed  can  only  last  about  eight 
minutes:  and  in  these  latitudes,  only  about  six  minutes. 

The  darkness  during  a  total  eclipse,  though  subject  to  some 
variation,  is  scarcely  so  intense  as  might  be  expected.  The  sky 
often  assumes  a  dusky,  livid  color,  and  terrestrial  objects  are 
similarly  affected.  The  brighter  planets  and  some  of  the  stars 
of  the  first  magnitude  generally  become  visible ;  and  sometimes 
etars  of  the  second  magnitude  are  seen.  The  corona  and  the 
rose-colored  protuberances  described  in  Art.  102  also  make 
their  appearance.  When  the  sun's  disc  has  been  reduced  to  a 
narrow  crescent,  it  sometimes  appears  as  a  succession  of  bright 
points,  separated  by  dark  spaces.  This  phenomenon  bears  the 
name  of  Baily's  beads.  The  dark  spaces  are  supposed  to  be 
the  lunar  mountains,  projected  upon  the  sun's  disc,  and  allowing 
the  disc  to  show  between  them. 

Occasionally  the  moon's  disc  is  faintly  seen,  shining  with  a 
dusky  light.  This  is  caused  by  the  rays  of  the  sun,  reflected 
back  to  the  moon  by  that  portion  of  the  earth's  surface  which 
is  still  illuminated  by  the  sun :  just  as  at  the  time  of  new  moon 
its  entire  disc  is  rendered  visible. 

CYCLE   AND   NUMBER   OF    ECLIPSES. 

162.  Cycle  of  Eclipses. — In  order  that  either  a  solar  or  a 
lunar  eclipse  shall  occur,  it  is  necessary,  as  we  have  seen,  that 
the  moon  shall  be  near  the  ecliptic  (in  other  words,  near  the 
line  of  nodes  of  its  orbit),  at  either  conjunction  or  opposition. 
It  is  evident  that  when  the  moon  is  near  the  line  of  nodes  at 
such  a  time,  the  sun  also  must  be  near  the  same  line.  The 


NUMBER   OF   ECLIPSES.  143 

occurrence  of  eclipses,  then,  depends  on  the  relative  situations 
of  the  sun,  the  moon,  and  the  moon's  nodes,  and  is  only  pos- 
sible when  they  are  all  in,  or  nearly  in,  the  same  straight  line. 
AVe  have  already  seen  (Art.  128)  that  the  line  of  nodes  is  con- 
tinually revolving  to  the  west,  completing  a  revolution  in  about 
18.6  years.  The  sun,  then,  in  its  apparent  path  in  the  ecliptic 
will  move  from  one  of  the  moon's  nodes  to  the  same  node  again 
in  less  than  a  year.  This  interval  of  time  may  be  called  the 
synodical  period  of  the  node,  and  is  found  to  be  346.62  days. 

Now,  we  have, 

19  X  346.62d.  =  6585.8d: 

and,  the  lunar  month  being  29.53  days,  we  have  also, 
223  X  29.53d.  =  6585.2d. 

If,  then,  the  moon  is  full  and  at  its  node  on  any  day,  it  will 
again  be  full,  and  at  the  same  node,  or  very  nearly  at  it,  after 
an  interval  of  6585  days :  and  the  eclipses  which  have  occurred 
in  that  interval  will  occur  again  in  very  nearly  the  same  order. 
This  period  of  6585  days,  or  18  years  and  10  days,  is  called  the 
cycle  of  eclipses.  It  was  known  to  the  Chaldsean  astronomers 
under  the  name  of  Saros.  Care  must  be  taken  not  to  confound 
this  cycle  with  the  lunar  cycle  described  in  Art.  147. 

163.  Number  of  Eclipses. — Since  the  limit  of  the  moon's  lati- 
tude is  greater  in  the  case  of  a  solar  eclipse  than  in  the  case 
of  a  lunar  eclipse,  there  are  more  solar  eclipses  than  lunar 
eclipses.  Usually  70  eclipses  occur  in  a  cycle,  of  which  41 
are  solar  and  29  are  lunar.  Since  we  know  that  a  solar  eclipse 
is  inevitable  when  the  moon  is  so  near  the  line  of  nodes  at 
conjunction  that  its  latitude  is  less  than  1°  23'  15",  we  can  com- 
pute the  corresponding  angular  distance  of  the  sun  at  the  same 
time  from  this  line ;  and  having  computed  this,  we  may  also 
determine  the  length  of  time  required  by  the  sun  in  passing 
through  double  this  angle,  or,  in  other  words,  the  time  required 
in  passing  from  one  of  these  limits  to  the  corresponding  limit  on 
the  other  side  of  the  same  node.  If  we  do  this,  we  shall  find 
that  the  sun  cannot  pass  either  node  of  the  moon's  orbit  without 
being  eclipsed :  and  therefore  there  must  be  at  least  two  solar 
eclipses  in  a  year.  The  greatest  number  that  can  occur  is  five. 
The  greatest  number  of  lunar  eclipses  in  the  year  is  three,  and 


144  OCCULTATIONS. 

there  may  be  none  at  all.     The  greatest  number  of  both  kinds 
of  eclipses  in  a  year  is  seven  ;  the  usual  number  is  four. 

Although  the  annual  number  of  solar  eclipses  throughout  the 
whole  earth  is  the  greater,  yet  at  any  one  place  more  lunar  eclipses 
are-visible  than  solar.  The  reason  of  this  is  that  a  lunar  eclipse, 
when  it  does  occur,  is  visible  over  an  entire  hemisphere,  while 
the  area  within  which  a  solar  eclipse  is  visible  is  very  much 
more  limited. 

OCCULTATIONS. 

164.  An  occultation  of  a  planet  or  a  star  will  occur  whenever 
the  planet  or  star  is  so  situated  in  latitude  as  to  allow  the  moon 
to  come  in  between  it  and  the  earth.  In  order  to  determine  the 
limit  of  a  planet's  latitude  within  which  an  occultation  of  the 
planet  is  possible,  let  us  refer  to  Fig.  61.  In  this  figure,  E  is  the. 

A 


Fig.  61. 

centre  of  the  earth,  P  that  of  a  planet,  and  M  that  of  the  moon. 
An  occultajion  will  occur  when  the  moon  comes  between  the 
tangent  lines  GB  and  AH.  Let  EC  be  the  plane  of  the  ecliptic. 
PEC  is  then  the  geocentric  latitude  of  the  planet,  and  MEG 
that  of  the  moon. 

We  have, 

PEC  =  PEG  +  GET)  +  DEM  +  MEC, 
and  also, 

GET)  =  EDB  —  EGD. 

Now,  PEG  is  the  planet's  semi-diameter,  EGD  its  horizontal 
parallax,  DEM  the  moon's  semi-diameter,  EDB  its  horizontal 
parallax,  and  MEC,  as  above  stated,  its  latitude.  The  value 


OCCULT  ATIONS.  ".  15 

of  PEC,  therefore,  can  very  readily  be  obtained.  If  P,  instead 
of  representing  a  planet,  represents  a  star,  the  distance  PE 
becomes  so  great  that  AH  and  BG  are  sensibly  parallel,  and 
the  star's  parallax  and  semi-diameter  reduce  to  zero.  In  this 
case  the  greatest  value  of  PEC,  within  which  an  occupation 
can  occur,  will  be  the  sum  of  5°  20'  6",  61'  32",  and  16'  45", 
which  is  6°  38'  23". 

Since  the  moon  moves  from  west  to  east,  the  occultation 
always  takes  place  at  its  eastern  limb.  From  new  moon  to  full, 
the  dark  portion  of  the  moon  is  to  the  east,  as  may  be  seen  in 
Fig.  54,  and  from  full  moon  to  new,  the  bright  limb  is  to  the 
east.  When  an  occultation  occurs  at  the  dark  edge,  particu- 
larly if  the  moon  is  so  far  on  towards  its  first  quarter  that  the 
dark  portion  is  invisible,  the  disappearance  is  extremely  strik- 
ing, as  the  occulted  body  appears  to  be  extinguished  without 
any  visible  interference. 

As  already  stated  in  Art.  83,  a  solar  eclipse,  or  an  occultation 
of  a  planet  or  star,  although  not  visible  at  different  places  at 
the  same  absolute  instant  of  time,  may  still  be  made  the  means 
of  very  accurately  determining  the  longitude  of  a  place,  or  the 
difference  of  longitude  of  two  places.  For  instance,  in  the  case 
of  a  solar  eclipse,  we  may  deduce,  from  the  local  times  of  the 
beginning  and  the  end  of  the  eclipse,  as  observed  at  any  place, 
the  time  of  true  conjunction  of  the  sun  and  the  moon:  the  time 
of  conjunction,  that  is  to  say,  as  seen  from  the  centre  of  the  earth. 
If,  then,  we  compare  the  local  time  of  true  conjunction  with  the 
Greenwich  time  of  true  conjunction,  it  amounts  to  comparing 
the  local  and  the  Greenwich  timje  corresponding  to  the  same 
absolute  instant:  and  the  difference  of  these  two  times  will  evi- 
dently be  the  longitude  of  the  place  of  observation  from  Green- 
wich. 

13 


146 


TIDES. 


CHAPTER  XL 

THE   TIDES. 

165.  THE  surface  of  the  ocean  rises  and  falls  twice  in  the 
course  of  a  lunar  day,  the  length  of  which  is,  as  we  have  already 
seen  (Art.  143),  about  24h.  50m.  of  mean  solar  time.     The  rise 
of  the  water  is  called  flood  tide,  and  the  fall  ebb  tide.     When  the 
water  is  at  its  greatest  height  it  is  said  to  be  high  water,  and 
when  at  its  least  height,  low  water. 

166.  Cause  of  the  Tides. — The  tides  are  due  to  the  inequality 
of  the  attractions  exerted  by  the  moon  upon  the  earth  and  the 
waters  of  the  ocean,  and  to  a  similar  but  smaller  inequality  in 
the  attractions  exerted  by  the  sun. 

In  order  to  examine  the  phenomena  of  the  tides,  we  will  con- 
sider the  earth  to  be  a  solid  globe,  surrounded  by  a  shell  of 
water  of  uniform  depth.  The  centrifugal  force  induced  by  the 
rotation  of  the  earth  would  tend  to  give  a  spheroidal  form  to 
this  shell  of  water;  but  as  we  wish  simply  to  examine  the  effects 
of  the  attractions  exerted  by  the  moon  and  the  sun,  we  will  dis- 
regard the  notation  of  the  earth,  and  consider  it  to  be  at  rest. 

In  Fig.  62,  then.  letABCD  represent  the  earth,  and  the  dotted 


Fig.  62. 


line  GHIKihe  surrounding  shell  of  water.    Let  Mloe  the  moon. 
The  attraction  of  the  moon  on  the  solid  mass  of  the  earth  is  the 


TIDES.  147 

same  that  it  would  be  if  the  whole  mass  were  concentrated  at 
the  point  E.  Now  since,  by  the  law  of  gravitation,  the  at- 
traction of  the  moon  on  any  two  particles  is  inversely  as  the 
square  of  the  distances  of  the  two  particles  from  the  moon,  the 
attraction  exerted  upon  the  particle  of  water  at  G  will  be  greater 
than  that  exerted  upon  the  general  mass  of  the  earth,  supposed 
to  be  concentrated  at  E.  The  particle  G  will  therefore  tend  to 
recede  from  the  earth :  that  is  to  say,  its  gravity  towards  the 
earth's  centre  will  be  diminished,  although,  as  is  plain,  it  will  not 
move.  The  same  result  will  follow  at  the  opposite  point  /. 
The  moon  will  exert  a  greater  attractive  power  upon  the  mass 
of  the  earth  than  upon  a  particle  at  /,  and  will  tend  to  draw  the 
earth  away  from  the  particle:  so  that  the  gravity  of  the  particle 
at  I  towards  the  earth's  centre  will  also  be  diminished.  Since 
the  ratio  of  the  distances  ME  and  MG  is  very  nearly  equal  to 
the  ratio  of  the  distances  MI  and  ME,  the  amount  of  the  decrease 
of  gravity  at  G  and  at  I  will  be  nearly  the  same. 

Let  us  next  examine  the  effect  of  the  moon's  attraction  at 
some  point  L,  not  situated  vertically  under  the  moon.  The  at- 
traction of  the  moon  at  this  point  is  less  than  that  at  the  point 
G,  since  the  distance  ML  is  greater  than  the  distance  MG;  and 
since  the  attraction  exerted  on  the  mass  of  the  earth  is,  of  course, 
the  same  for  both  points,  the  difference  of  the  attractions  exerted 
on  the  earth  and  the  water  is  less  at  the  point  L  than  at  the 
point  G.  At  the  point  L,  however,  this  inequality  of  attraction 
is  not  wholly  counteracted  by  gravity :  for  if  the  force  with  which 
the  moon  tends  to  draw  a  particle  at  L  along  the  line  ML  be  re- 
solved into  two  forces,  one  in  the  direction  of  the  radius  EL, 
and  the  other  in  the  direction  of  the  tangent  LT,  the  latter  force 
will  cause  the  particle  to  move  towards  the  point  G.  The  same 
result  will  follow  at  any  other  point  of  the  arc  HGK:  so  that 
all  the  water  in  that  arc  tends  to  flow  towards  the  point  G,  and 
to  produce  high  water  there. 

In  the  same  way  it  may  be  shown  that  the  water  in  the  arc 
HIK  tends  to  flow  towards  the  point  /,  and  to  produce  high 
water  at  that  point. 

The  result,  then,  of  the  attraction  of  the  moon,  exerted  under 
the  suppositions  which  we  made  at  the  outset,  is  to  give  to  the  shell 


148 


TIUES. 


N 


of  water  a  spheroidal  form,  as  shown  in  the  figure,  the  major 
axis  of  the  spheroid  being  directed  towards  the  moon.  Suitable 
investigation  shows  that  the  difference  of  the  major  and  the  minor 
semi-axis  of  this  spheroid  is  about  fifty-eight  inches. 

167.  Daily  Inequality  of  the  Tides. — The  rotation  of  the  earth, 
and  the  inclination  of  the  plane  of  the  moon's  orbit  to  the  plane 
of  the  equator,  produce  in  general  an  inequality  in  the  two  daily 
tides  at  any  place.  In  order  to  show  this,  we  will  suppose  that 
the  spheroidal  form  of  the  water  is  assumed  instantaneously  in 
each  new  position  of  the  earth  as  it  rotates.  In  Fig.  63,  let  E 

M  be  the  centre  of  the 
earth,  surrounded, 
as  in  Fig.  62,  by 
a  spheroidal  shell 
of  water,  the  trans- 
verse axis  of  the 
spheroid  lying  in 
the  direction  of  the 
moon  M.  Let  Pp 
be  the  axis  of  rota- 
tion of  the  earth, 
and  CD  the  equator. 
The  angle  MED  is 
the  moon's  declination.  The  water  is  at  its  greatest  height,  as  be- 
fore, at  the  points  A  and  B,  and  the  height  at  other  points  dimi- 
nishes as  the  angular  distance  of  those  points  from  the  line  GH 
increases.  Let  /  be  a  place  having  the  same  latitude  that  A  has, 
but  situated  180°  from  it  in  longitude.  The  height  of  the  tide 
at  /is  represented  by  IK.  In  a  little  more  than  twelve  hours 
the  rotation  of  the  earth  will  have  caused  I  and  A  to  change 
places  with  reference  to  the  moon.  I  will  then  be  where  A  is 
in  the  figure,  and  will  have  a  tide  with  the  height  A  G,  while 
A  will  be  where  /is  now,  and  will  have  a  tide  with  the  height 
IK.  It  is  not  necessary  to  prove  that  IK  is  less  than  A  G.  We 
see,  then,  that  at  both  A  and  /  the  two  daily  tides  are  unequal, 
the  greater  of  the  two  occurring  at  each  place  at  the  time  of  the 
moon's  upper  culmination  at  that  place,  or  being,  at  all  events, 
the  one  which  occurs  next  after  that  culmination.  The  same 


Fig.  63. 


TIDES.  149 

daily  inequality  of  tides  may  be  shown  to  exist  at  any  other 
points  on  the  earth's  surface,  as,  for  instance,  at  L  and  0.  At 
the  equator,  however,  and  also  at  the  poles,  the  two  daily  tides 
are  sensibly  equal,  as  may  readily  be  seen  from  the  figure. 

168.  General  Laws. — As  far  as  the  influence  of  the  moon  is 
concerned  in  causing  tides,  the  following  general  laws  may  be 
deduced  from  what  has  been  shown  in  the  preceding  articles. 

(1.)  When  the  moon  is  in  the  plane  of  the  celestial  equator,  or, 
in  Fig.  63,  when  EM  coincides  with  ED,  the  tides  are  greatest 
at  the  equator,  and  diminish  at  other  places  as  the  latitude  in- 
creases ;  and  the  two  daily  tides  at  any  place  are  sensibly  equal. 

(2.)  When  the  moon  is  not  in  the  plane  of  the  celestial  equa- 
tor, the  two  daily  tides  at  any  place  except  the  poles  and  the 
equator  are  unequal.  The  greatest  tides,  and  the  greatest  ine- 
quality of  tides,  occur  at  those  places  whose  latitude  is  numeri- 
cally equal  to  the  moon's  declination.  If  the  place  is  on  the 
same  side  of  the  equator  as  the  moon,  the  greater  of  the  two 
daily  tides  occurs  at  or  next  after  the  upper  culmination  of  the 
moon;  if  the  place  is  on  the  opposite  side  of  the  equator,  the 
greater  tide  occurs  at  or  next  after  the  lower  culmination  of  the 
moon. 

(3.)  Owing  to  the  retardation  of  the  moon  (Art.  143),  there  is 
a  similar  retardation  in  the  occurrence  of  high  water.  The 
length  of  the  lunar  day  being  on  the  average  24h.  50m.,  the  aver- 
age interval  of  time  between  two  successive  tides  is  12h.  25m. 

169.  Influence  of  the  Sun  in  Causing  Tides. — All  that  has  been 
said  in  the  preceding  articles  with  regard  to  the  influence  of  the 
moon  in  creating  tides  is  equally  true  with  regard  to  the  influ- 
ence of  the  sun.     The  mass  of  the  sun  being  so  immense  in 
comparison  with  that  of  the  moon,  it  might  be  supposed  that 
the  influence  of  the  sun  over  the  tides  would  be  greater  than 
that  of  the  moon,  even  although  its  distance  from  the  earth  is 
much  greater  than  that  of  the  moon.     But  such  is  not  the  case 
in  fact.     The  height  of  the  tide  produced  by  either  body  is  not 
so  much  due  to  the  absolute  attraction  which  that  body  exerts, 
as  to  the  relative  attractions  which  it  exerts  on  the  solid  mass  of 
the  earth  and  on  the  water :  and  the  moon  is  so  much  nearer  to 
the  earth  than  the  sun,  that  the  difference  of  its  attractions  on 


150  PRIMING    AND    LAGGING. 

the  earth  and  the  water  is  greatej  than  the  corresponding  dif- 
ference in  the  case  of  the  sun.  It  is  computed  that  the  effect 
of  the  moon  in  creating  tides  is  about  2i  times  that  of  the  sun. 

170.  Combined  Effects  of  both  Sun  and   Moon. — Since   each 
body,  independently  of  the  other,  tends  to  raise  the  surface  of  the 
water  at  certain  points,  and  to  depress  it  at  certain  other  points, 
the  tides  will  evidently  be  higher  when  both  bodies  tend  to  raise 
the  surface  of  the  water  at  the  same  time,  than  when  one  tends 
to  raise  and  the  other  to  depress  it.     At  new  and  full  moon  the 
two  bodies  act  together,  while  at  the  first  and  the  last  quarter 
they  act  in  opposition  to  each  other.     The  tides  at  the  former 
periods  will  therefore  be  the  greater,  and  are  called  spring  tides ; 
and  the  tides  at  the  latter  periods  are  called  neap  tides.    The  ratio 
of  the  spring  to  the  neap  tide  is  that  of  (2i  -f-  1)  to  (2£  — 1), 
or  of  5  to  2. 

The  height  of  the  tide  is  also  affected  by  the  change  in  the 
distance  of  the  attracting  body.  For  instance,  when  the  moon 
is  in  perigee,  the  tides  tend  to  run  higher  than  when  the  moon  is  in 
apogee;  and  when  the  moon  is  in  perigee,  and  also  either  new  or 
full,  unusually  high  tides  will  occur. 

171.  Priming  and  Lagging  of  the  Tides. — Each  of  these  bodies 
may  be  supposed  to  raise  a  tidal  wave  of  its  own,  and  the  actual 
high  water  at  any  place  may  be  considered  to  be  the  result  of 
the  combination  of  the  two  waves.     When  the  moon  is  in  its 
first  or  its  third  quarter,  the  solar  wave  is  to  the  west  of  the  lunar 
one,  and  the  actual  high  water  will  be  to  the  west  of  the  place 
at  which  it  would  have  been  had  the  moon  acted  alone.     There 
is  therefore  at  these  times  an  acceleration  of  the  time  of  high  water, 
which  is  called  the  priming  of  the  tides.     In  the  second  and  the 
fourth  quarter,  the  solar  wave  is  to  the  east  of  the  lunar  one, 
and  a  retardation  of  the  time  of  high  water  occurs,  which  is 
called  the  lagging  of  the  tides. 

172.  Although  the  theory  of  the  tides,  on  the  supposition  that 
the  earth  is  wholly  covered  with  water,  admits  of  easy  explana- 
tion, the  actual  phenomena  which  they  present  are  very  much 
more  complicated,  and  must  be  obtained  principally  from  obser- 
vation.    The  lunar  wave  mentioned  in  the   preceding  article 
being  greater  than  the  solar  wave,  we  may  consider  the  two  to- 


ESTABLISHMENT.  151 

gether  to  constitute  one  great  tidal  wave,  which  at  every  moment 
tends  to  accompany  the  moon  in  its  apparent  diurnal  path  to- 
wards the  west,  raising  the  waters  at  successive  meridians,  but 
giving  them  little  or  no  progressive  motion.  This  tidal  wave 
would  naturally  move  westward  with  an  angular  velocity  equal 
to  that  of  the  moon,  so  that  at  the  equator  its  motion  would  he 
about  1000  miles  an  hour ;  but  the  obstructions  offered  by  the 
continents,  the  irregularity  of  their  outlines,  the  uneven  surface 
of  the  ocean  bed,  and  the  action  of  winds  and  currents  and  fric- 
tion, all  combine  not  only  to  diminish  the  velocity  of  the  tidal 
wave,  but  also  to  make  it  extremely  variable. 

173.  Establishment  of  a  Port. — The  interval  of  time  between 
the  moon's  transit  over  any  meridian  and  high  water  at  that 
meridian  varies  at  different  places,  and  varies  also  on  different 
days  at  the  same  place.  This  interval  of  time  is  called  the 
luni-tidal  interval.  The  mean  of  the  values  of  this  interval  on 
the  days  of  new  and  full  moon  is  called  the  common  establish- 
ment of  a  port.  The  mean  of  all  the  luni-tidal  intervals  in  the 
course  of  the  month  is  called  the  corrected  establishment.  These 
establishments  are  obtained  by  observation,  and  are  given  in 
Bowditch's  Navigator,  and  also  in  other  works.  Thus  the  estab- 
lishment of  Annapolis  is  4h.  38m.,  and  that  of  Boston  llh.  12m. 
The  time  of  transit  of  the  moon  over  any  meridian  on  any  day 
can  be  obtained  from  the  Nautical  Almanac:  and  the  sum  of 
this  time  of  transit  and  of  the  establishment  of  any  port  will  be 
the  approximate  time  of  that  flood  tide  which  occurs  next  after 
the  transit.  Suppose,  for  instance,  the  time  of  high  water  at 
Annapolis,  on  January  11,  1867,  is  desired.  We  have  from  the 
Almanac  the  time  of  the  moon's  transit,  4h.  33m. ;  adding  to  this 
the  establishment,  4h.  38m.,  the  sum  is  9h.  llm.  This  is,  in  this 
case,  the  time  of  the  evening  tide.  The  time  of  the  morning  tide 
may  be  obtained  by  subtracting  12h.  25m.  from  this  time,  which 
will  give  us  8h.  46m.  A.M.  A  more  accurate  result  in  this 
last  case  might  be  obtained  by  taking  from  the  Almanac  the 
time  of  the  preceding  lower  transit,  and  adding  the  establish- 
ment to  it;  but  practically  this  would  be  a  needless  refinement, 
for  the  two  results  would  only  vary  by  about  two  minutes. 

The  time  of  transit  which  the  Almanac  gives  is  in  astronomical 


152  COTIDAL    LINES. 

time:  hence  the  resulting  time  of  J^igh  water  will  also  be  in  astro- 
nomical time,  and  it  will  frequently  happen  that  the  time  which 
we  find,  when  turned  into  civil  time,  will  fall  on  the  civil  day 
subsequent  to  the  day  for  which  the  time  is  desired.  Take,  for 
instance,  January  23, 1867,  at  Annapolis.  The  time  of  transit  is 
January  23,  15h.  34m. :  hence  the  time  of  high  water  is  Janu- 
ary 23,  20h.  12m.,  or,  in  civil  time,  January  24,  8h.  12m.  A.M. 
If,  then,  we  wish  the  time  of  high  water  on  the  morning  of  the 
civil  day  January  23d,  we  must  take  from  the  Almanac  the  time 
of  transit  for  the  astronomical  day  of  January  22d. 

There  are  other  tables  given  in  Bowditch's  Navigator,  and  in 
the  United  States  Coast  Survey  Reports,  by  which  the  time  of 
high  water  can  be  obtained  with  greater  accuracy  than  by  the 
method  given  above. 

174.  Cotidal  Lines. — If  the  tidal  wave  were  everywhere  uni- 
form in  its  progress,  it  would  come  to  all  points  on  the  same 
meridian  at  the  same  time.    But,  owing  to  irregularities  induced 
by  local  causes,  such  is  not  the  case,  and  places  on  different 
meridians  often  have  high  water  at  the  same  instant  of  time. 
Charts  are  therefore  published  on  which  are  drawn  lines  con- 
necting places  where  high  tides  occur  at  the  same  instant:  and 
these  lines  are  called  cotidal  lines.     These  lines  are  usually  ac- 
companied by  numerals,  which  indicate  the  hours  of  Greenwich 
time  at  which  high  tides  occur  on  the  days  of  new  and  full  moon 
along  the  different  lines. 

175.  Height  of  Tides. — At  small  islands  in  mid-ocean  the  height 
of  the  tides  is  not  great,  being  sometimes  less  than  one  foot.  When 
the  tidal  wave  approaches  a  continent,  and  the  water  begins  to 
shoal,  the  velocity  of  the  wave  is  diminished,  and  the  height  of 
the  tide  is  increased.     When  the  wave  enters  bays  opening  in 
the  direction  in  which  the  wave  is  moving,  the  height  of  the  tide 
is  still  further  increased. 

The  eastern  coast  of  the  United  States  may  be  considered  to 
constitute  three  great  bays :  the  first  included  between  Cape  Sable, 
in  Nova  Scotia,  and  Nantucket,  the  second  between  Nantucket 
and  Cape  Hatteras,  and  the  third  between  Cape  Hatteras  and 
Cape  Sable,  in  Florida.  In  each  of  these  bays  the  tides,  in  gene- 
ral, increase  in  height  from  the  entrance  of  the  bay  to  its  head. 


HEIGHT    OF    TIDES.  153 

For  instance,  in  the  most  southern  of  these  bays,  the  tides  at 
Cape  Sable  and  Cape  Hatteras  are  not  more  than  two  feet  in 
height;  while  at  Port  Royal,  at  the  head  of  this  bay,  they  are 
about  seven  feet.  The  same  thing  is  noticed,  in  general,  in 
smaller  bays  and  sounds.  For  instance,  in  Long  Island  Sound, 
the  height  of  the  tide  is  two  feet  at  the  eastern  extremity,  and 
more  than  seven  feet  at  the  western.  This  increase  of  height  is 
particularly  noticeable  in  the  Bay  of  Fundy,  in  which  the  height 
is  eighteen  feet  at  the  entrance,  and  fifty  and  sometimes  seventy 
feet  at  the  head. 

There  are  in  some  cases,  however,  special  causes  which  create 
exceptions  to  this  general  rule  of  increase  of  the  tides  between 
the  entrance  and  the  head  of  a  bay.  In  Chesapeake  Bay,  for 
instance,  which  is  wider  at  some  places  than  it  is  at  the  entrance, 
and  which  lies  about  north  and  south,  the  tides  in  general  dimin- 
ish in  height  as  we  ascend  the  bay. 

176.  Tides  in  Rivers. — The  same  general  principle  holds  good 
in  the  tides  of  rivers.     When  the  channel  contracts  or  shoals 
rapidly,  the  height  of  the  tide  increases:    when  it  widens  or 
deepens,  the  height  decreases.     In  a  long  river,  then,  the  tides 
may  alternately  increase  and  decrease.     For  instance,  at  Tivoli, 
on  the  Hudson,  between  West  Point  and  Albany,  the  tide  is 
higher  than  it  is  at  either  of  those  two  places. 

177.  Different  Directions  of  the  Tidal  Wave. — The  tidal  wave 
naturally  tends  to  move  towards  the  west ;  but  the  obstructions 
offered  by  the  continents  and  the  promontories,  and  the  irregular 
conformation  of  the  bottom  of  the  ocean,  materially  change  the 
direction  of  its  motion.     Sometimes  its  direction  is  even  towards 
the  east.     From  a  point  about  one  thousand  miles  southwest  of 
South  America  there  appear  to  start  two  tidal  waves,  which  travel 
in  nearly  opposite  directions,  one  towards  the  west  and  the  other 
towards  the  east. 

178.  Four  Daily  Tides. — At  some  places  the  tides  rise  and 
fall  four  times  in  each  day.     This  is  ascribed  to  the  existence  of 
two  different  tidal  waves,  coming  from  opposite  directions.    This 
phenomenon  occurs  on  the  eastern  coast  of  Scotland,  where  one 
wave  comes  into  the  North  Sea  through  the  English  Channel, 
while  a  second  wave  comes  in  around  the  northern  extremity  of 


154  TIDES   IN    LAKES. 

Scotland.     At  places  where  these  two  waves  arrive  at  different 
times,  each  wave  will  produce  two  daily  tides. 

179.  Tides  in  Lakes  and  Inland  Seas.  —  If  there  is  any  tide  in 
lakes  and  inland  seas,  it  is  usually  so  slight  as  to  be  scarcely 
measurable.     A  series  of  careful  observations  has  demonstrated 
the  existence  of  a  tide  in  Lake  Michigan,  which  is  at  its  height 
about  half  an  hour  after  the  moon's  transit.  The  average  height 
which  it  attains,  however,  is  less  than  two  inches. 

180.  Other  Phenomena.  —  Along  the  northern  coast  of  the  Gulf 
of  Mexico  there  is  only  one  tide  in  the  day,  the  second  one  being 
probably  obliterated  by  the  interference  of  two  waves.     An  ap- 
proximation to  this  state  of  things  is  noticed  on  the  Pacific  coast, 
where  at  times  one  of  the  daily  tides  has  a  height  of  several  feet, 
and  the  other  a  height  of  only  a  few  inches.     A  very  curious 
statement  is  made  by  missionaries  in  reference  to  the  tides  at  the 
Society  Islands.     They  say  that  the  tides  there  are  uniform,  not 
only  in  the  height  which  they  attain,  but  in  the  time  of  ebb  and 
flow,  high  tide  occurring  invariably  at  noon  and  at  midnight  :  so 
that  the  natives  distinguish  the  hour  of  the  day  by  terms  de- 
Bcriptive  of  the  condition  of  the  tide. 


It  is  now  generally  admitted  that  one  result  of  the  friction  of 
the  tides  is  a  diminution  of  the  velocity  of  the  earth's  rotation  ;  and  it  is 
possible  that  the  moon's  secular  acceleration  (page  133)  is  partly  due,  not 
to  an  increase  in  the  moon's  orbital  velocity,  but  to  this  same  diminution 
of  the  earth's  rotation.  The  amount  of  the  diminution  is,  however,  so  very 
small  that  all  attempts  to  compute  it  have  been  thus  far  unsuccessful. 


PLANETS.  15.r) 


CHAPTER  XII. 

THE   PLANETS   AND   THE   PLANETOIDS.      THE   NEBULAR 
HYPOTHESIS. 

181.  The  Planets  and  their  Apparent  Motions. — There  are  other 
celestial  bodies  besides  the  sun  and  the  moon,  which,  while  they 
share  the  common  diurnal  motion  towards  the  west,  appear  to 
change  their  relative  positions  among  the  stars.  These  bodies  are 
called  planets,  from  a  Greek  word  signifying  wanderer.  Some  of 
them  are  visible  to  the  naked  eye,  and  some  only  become  visible 
by  the  aid  of  a  telescope.  In  some  of  them  the  change  of  position 
among  the  stars  becomes  apparent  from  the  observations  of  a 
few  nights :  while  in  others  even  the  annual  change  of  position 
is  so  small  that  they  have  for  ages  been  considered  to  be  fixed 
stars.* 

This  change  of  position  is  determined,  as  it  was  determined  in 
the  case  of  the  sun  and  the  moon,  by  observations  of  their  right 
ascensions  and  declinations.  When  such  observations  are  made, 
the  apparent  motions  of  the  planets  are  found  to  be  very  irregular. 
Sometimes  they  appear  to  move  towards  the  east,  and  sometimes 
towards  the  west,  while  at  other  times  they  appear  to  be  station- 
ary in  the  heavens.  Such  irregularity  in  the  direction  of  their 
motion  is  at  once  seen  to  be  incompatible  with  the  supposition 

*  The  times  of  meridian  passage  of  all  the  planets  which  ever  become 
conspicuously  visible  are  given  in  the  American  Ephemeris.  The  altitude 
which  any  planet  has  when  on  the  meridian  is  obtained  from  the  corre- 
sponding zenith  distance,  and  this  is  found  at  any  pla^ce  whose  latitude 
is  known,  from  the  formula 

*=:£—  d: 

the  declination  being  also  given  in  the  Ephemeris.  It  must  be  remembered 
that  (L  —  d)  becomes  numerically  (Zr  +  d)  when  L  arid  d  have  different 
names ;  that  z  in  any  case  has  the  same  name  as  (L  —  d) ;  and  that  when  a 
is  north,  the  bearing  of  the  body  is  south,  a.nd  conversely. 


156  PLANETS. 

that  they  are,  like  the  moon,  satellites  of  the  earth,  revolving 
about  it  as  a  centre.  The  next  supposition  that  will  naturally 
be  made  is  that  they  may  revolve  about  the  sun. 

182.  Heliocentric  Parallax. — In  order  to  test  the  correctness 
of  this  second  supposition,  we  must  first,  from  the  apparent 
motions  which  are  observed  from  the  earth,  deduce  the  corre- 
sponding motions  which  would  be  seen  by  an  observer  stationed 
at  the  sun.  Fig.  64  will  serve  to  show  how,  by  means  of  a  body's 
heliocentric  parallax  (Art.  56),  the  position  which  that  body 
would  have  if  seen  from  the  sun's  centre — in  other  words,  its  helio- 
centric position — may  be  determined  from  its  geocentric  position. 
In  this  figure  let  S  represent  the  sun,  AB  CE  the  earth's  orbit, 
the  plane  of  which  intersects  the  celestial  sphere  in  the  circle 


Fig.  64. 

,  and  P  the  position  of  a  planet  when  projected  upon  the 
plane  of  the  ecliptic.  Let  the  vernal  equinox  be  supposed  to  lie 
in  the  direction  EV  or  SV,  which  two  lines  must  be  supposed 
sensibly  to  meet  when  prolonged  to  the  celestial  sphere.  Draw 
the  lines  EP  and  SP.  Since  the  distances  of  the  planet  from  the 
sun  and  the  earth  are  finite,  these  lines  will  not  lie  in  the  same 
direction,  and  the  angle  EPS  which  they  make  with  each  other 
will  be  the  difference  of  the  directions  in  which  the  planet  is  seen 
from  the  earth  and  the  sun  :  in  other  words,  the  planet's  helio- 


ORBITS    OF    THE   PLANETS.  157 

centric  parallax  in  longitude.  Through  S  draw  SK  parallel  to 
ED.  The  angle  VEP,  or  its  equal,  VSK,  is  the  planet's  geo- 
centric longitude,  and  is  obtained  by  observation.  The  sum  of 
this  angle  and  the  angle  EPS  is  the  angle  VSP,  or  the  planet's 
heliocentric  longitude.  Provided,  then,  we  know  the  angle  EPS, 
we  can  readily  obtain  the  angle  VSP.  Now,  in  the  triangle  PES 
we  know  from  Kepler's  Third  Law  the  ratio  of  the  sides  SP 
and  ES,  which  are  the  distances  of  the  planet  and  the  earth  from 
the  sun;  and  the  angle  PES,  the  planet's  angular  distance  from 
the  sun,  or  its  elongation  (Art.  56),  can  be  obtained  by  observa- 
tion. The  angle  EPS  may  then  be  readily  computed. 

By  a  similar  method  the  planet's  heliocentric  latitude  may  be 
determined  from  its  geocentric  latitude,  and  the  heliocentric 
place  of  a  planet  may  thus  be  obtained  at  any  time. 

183.  Orbits  of  the  Planets. — When  the  motions  of  the  planets 
as  they  would  be  seen  from  the  centre  of  the  sun  are  thus  deduced 
from  their  observed  motions  with  reference  to  the  earth,  all  the 
apparent  irregularities  of  motion  disappear.     The  planets  are 
found  to  revolve  from  west  to  east  in  ellipses  about  the  sun  in 
one  of  the  foci,  the  eccentricity  of  the  ellipses  diminishing,  as  a 
general  rule,  as  the  magnitude  increases.      The  planes  of  the 
orbits  are  found  to  be  nearly  coincident  with  the  plane  of  the 
ecliptic,  and  Kepler's  and  Newton's  laws  are  exactly  fulfilled  in 
the  case  of  each  planet.     The  line  in  which  the  plane  of  each 
planet  intersects  the  plane  of  the  ecliptic  is  called  the  line  of 
nodes,  and  the  terms  perihelion  and  aphelion  have  the  same 
signification  that  they  have  in  the  case  of  the  earth. 

184.  Inferior  and  Superior  Planets. — The  planets  are  divided 
into  two  classes :    inferior  and  superior  planets.     The  inferior 
planets  are  those  whose  distances  from  the  sun  are  less  than  the 
distance  of  the  earth  from  the  sun,  and  whose  orbits  are  therefore 
included  within  the  orbit  of  the  earth.     The  inferior  planets  are 
Mercury  and  Venus.*     The  superior  planets  are  those  whose 

*  There  are  some  astronomers  who  are  inclined  to  suspect  the  existence 
of  a  third  inferior  planet,  Vulcan,  distant  about  13,000,000  miles  from  the 
sun.     Two  American  observers  are  convinced  that  they  saw  such  a  planet 
during  the  total  solar  eclipse  of  Julv  29,  1878. 
14 


158 


INFERIOR    PLANETS. 


distances  from  the  sun  are  greater  than  that  of  the  earth,  and 
whose  orbits  therefore  include  the  orbit  of  the  earth.  The 
superior  planets  are  Mars,  Jupiter,  Saturn,  Uranus,  and  Neptune. 
There  is  besides  these  a  group  of  small  planets,  called  minor 
planets,  planetoids,  or  asteroids,  situated  between  Mars  and 
Jupiter.  Up  to  October,  1878,  192  of  these  minor  planets  had 
been  discovered.  The  earth  is  also  a  planet,  lying  between 
Venus  and  Mars.  It  is  therefore  a  superior  planet  to  Mercury 
and  Venus,  and  an  inferior  planet  to  the  other  planets.  Its 
sidereal  period  is  greater  than  the  periods  of  the  inferior  planets, 
and  less  than  those  of  the  superior  planets. 


INFERIOR   PLANETS. 

185.  In  Fig.  65  let  S  represent  the  sun,  pp'p"p""  the  orbit 
of  an  inferior  planet,  the  plane  of  which  is  supposed  to  coincide 


Fig.  65. 

with  the  plane  of  the  ecliptic,  ABDEt\\Q  orbit  of  the  earth,  and 
the  circle  KGH  the  intersection  of  the  plane  of  the  ecliptic  with 
the  celestial  sphere.  Suppose  the  earth  to  be  at  E.  When  the 
planet  is  at  P,  between  the  earth  and  the  sun,  or  at  P"'9  on  the 
opposite  side  of  the  sun  to  the  earth,  it  has  the  same  geocentric 


DIRECT   AND    RETROGRADE   MOTION.  159 

longitude  as  the  sun,  and  is  in  conjunction  with  it.  The  position 
at  P  is  called  the  inferior  conjunction,  and  that  at  P'"  the  superior 
conjunction. 

The  greatest  angular  distance  of  the  planet  from  the  sun  will 
evidently  occur  when  the  line  connecting  the  planet  and  the  earth 
is  tangent  to  the  orbit  of  the  planet ;  that  is  to  say,  when  the  planet 
is  at  P"  or  P"".  The  position  at  P"  is  called  the  greatest  western 
elongation  of  the  planet,  that  at  P""  the  greatest  eastern  elongation 
of  the  planet.  We  have  already  seen  in  Art.  92  that  the  rela 
tive  distances  of  the  planet  and  the  earth  from  the  sun  are  at 
once  obtained  when  we  have  measured  the  greatest  elongation. 

186.  Direct  and  Retrograde  Motion. — The  motion  of  the  infe- 
rior planets  is  always  in  reality  from  west  to  east,  or  direct,  as  it 
is  called;  but  when  the  planet  is  near  its  inferior  conjunction, 
its  motion  is  apparently  from  east  to  west,  or  retrograde.     This 
apparent  retrograde  motion  is  explained  in  Fig.  65.     Let  the 
planet  be  at  its  inferior  conjunction  at  P,  and  let  both  the  earth 
and  the  planet  move  on  about  the  sun  in  the  direction  EABD. 
The  angular  and  the  linear  velocity  of  the  planet  about  the  sun 
being  greater  than  they  are  in  the  case  of  the  earth,  when  the 
earth  arrives  at  E',  the  planet  will  be  at  some  point  P',  and  will 
lie  in  the  direction  E'G;  the  sun,  on  the  other  hand,  will  lie  in 
the  direction  E'S'.    While,  then,  the  earth  is  advancing  from  E 
to  E',  the  sun  and  the  planet  appear  to  move  away  in  opposite 
directions  from  the  point  (7,  on  which  both  were  projected  when  the 
earth  was  at  E.     But  the  apparent  motion  of  the  sun  is  invariably 
towards  the  east ;  hence  the  planet  has  apparently  moved  tow'ards 
the  west.     It  may  also  be  shown  that  the  same  apparent  retro- 
grade motion  occurs  when  the  planet  is  approaching  inferior 
conjunction,  and  is  within  a  short  distance  of  it. 

187.  Stationary   Points.— When   the   planet   is   at  P"",  it  is 
moving  directly  towards  the  earth  in  the  direction  P""E,  and 
the  motion  of  the  earth  in  its  orbit  gives  the  planet  an  apparent 
motion  in  advance.     The  same  must  also  be  the  case  when  the 
planet  is  at  P".     Since  then  the  motion  of  the  planet  is  direct 
at  the  greatest  elongations,  and  retrograde  at  inferior  conjunction, 
there -must  be  a  point  in  the  orbit  between  inferior  conjunction 
and  each  elongation  at  which  the  planet  neither  advances  nor 


160  ELEMENTS    OF    A    PLANET'S    ORBIT. 

recedes,  but  appears  stationary  4n  the  heavens.     These  points 
are  called  the  stationary  points. 

At  all  other  parts  of  the  orbit  except  those  which  have  been 
discussed,  the  apparent  motion  of  the  planet  is  direct ;  but  the 
velocity  with  which  it  moves  is  subject  to  great  variation.  It 
was  on  account  of  this  irregularity,  both  in  the  direction  and  the 
amount  of  their  apparent  motions,  that  these  bodies  were  called 
wandering  stars  by  the  ancient  Greeks. 

188.  Evening   and  Morning  Stars. — Except  at  the  times  of 
conjunction,  an  inferior  planet  is  either  to  the  east  or  to  the  west 
of  the  sun.     When  it  is  to  the  east  of  the  sun  it  will  set  after 
the  sun  has  set,  and  when  it  is  to  the  west  of  the  sun  it  will  rise 
before  the  sun  has  risen.     In  certain  parts  of  its  orbit  the  planet's 
elongation  from  the  sun  is  sufficiently  great  to  carry  the  planet 
beyond  the  limits  of  twilight,  or  18°  (Art.  101);  it  will  then  be 
an  evening  star  if  to  the  east  of  the  sun,  and  a  morning  star 
if  to  the  west.     It  is  only  to  Venus,  however,  that  these  terms 
are  commonly  applied,  at  least  so  far  as  the  inferior  planets  are 
concerned :  since  Mercury  is  so  near  to  the  sun  that  it  is  seldom 
visible,  and  even  when  it  is  visible,  it  appears  like  a  star  of  only 
the  third  or  the  fourth  magnitude. 

189.  Elements  of  a  Planet's  Orbit. — In  order  to  compute  the 
position  in  space  at  any  time  of  either  an  inferior  or  a  superior 
planet,  we  must  be  able  to  determine: — 

1st.  The  relative  position  of  the  plane  of  the  planet's  orbit  to 
the  plane  of  the  ecliptic ; 

2d.  The  position  of  the  orbit  itself  in  the  plane  in  which  it 
lies; 

3d.  The  magnitude  and  the  form  of  the  orbit;  and 

4th.  The  position  of  the  planet  in  its  orbit. 

These  four  conditions  require  the  knowledge  of  seven  distinct 
quantities.  The  first  condition  is  satisfied  if  we  know  (1)  the 
position  of  the  line  in  which  the  plane  of  the  orbit  intersects  the 
plane  of  the  ecliptic,  or,  what  amounts  to  the  same  thing,  the 
longitude  of  the  planet's  nodes,  and  (2)  the  inclination  of  the 
two  planes  to  each  other.  The  second  condition  is  satisfied  if 
we  know  (3)  th°  longitude  of  the  perihelion.  The  third  condi- 
tion is  satisfied  if  we  know  (4)  the  semi-major  axis,  or  the 


LONGITUDE   OF   THE  NODE. 


161 


]>lauet's  mean  distance  from  the  sun,  and  (5)  the  eccentricity  of 
the  orbit.  Finally,  the  last  condition  is  satisfied  if  we  know 
(6)  the  time  in  which  it  makes  one  complete  revolution  about 
the  sun,  or  its  periodic  time)  and  (7)  the  time  when  the  planet 
is  at  some  known  place  in  its  orbit,  as,  for  instance,  the  peri- 
helion. These  seven  quantities  are  called  the  elements  of  the  orbit. 
190.  Heliocentric  Longitude  of  the  Node. — A  planet  is  at  its 
nodes  when  its  latitude  is  zero;  and  if  the  heliocentric  longitude 
of  the  planet  at  that  time  can  be  determined,  it  will  also  be  the 
heliocentric  longitude  of  the  node,  since  the  line  of  nodes  of 
every  planet  passes  through  the  sun.  But  the  heliocentric  longi- 
tude of  a  planet  when  at  its  node  differs  from  the  geocentric 
longitude  (which  may  be  obtained  directly  from  observation), 
excepting  only  in  case  the  earth  itself  happens  at  that  time  to 
be  on  the  line  of  nodes.  .  We  must  therefore  be  able  to  deduce 
the  heliocentric  longitude  of  the  planet  from  the  geocentric. 
When  the  planet's  distance  from  the  sun  is  known,  this  can  be 
done  by  the  method  explained  in  Art.  182;  and  when  this  dis- 
tance is  not  known,  the  following  method  can  be  used. 

N 


In  Fig.  66,  let  8  be  the  sun,  POHK  the  orbit  of  a  planet, 
CDEE'  the  orbit  of  the  earth,  and  NM  the  line  of  nodes  of  the 


162  INCLINATION   OF   A   PLANET'S   ORBIT. 

planet.  Let  the  vernal  equinox  lie  in  the  direction  EV  or  SV. 
Let  E  be  the  position  of  the  earth  when  the  planet  is  on  the  line 
of  nodes  at  P.  The  elongation  of  the  planet,  or  the  angle  PES, 
and  the  geocentric  longitude  of  both  planet  and  node,  or  the 
angle  VEP,  can  be  obtained  by  observation.  Suppose  both 
planet  and  earth  to  move  on  in  their  orbits,  and  the  earth  to  be 
at  E'  when  the  planet  again  reaches  the  same  node,  and  let  the 
planet's  elongation  at  this  time,  or  the  angle  SE'P,  be  observed. 
Now,  since  the  earth's  orbit,  although  represented  in  the  figure 
by  a  circle,  is  raally  an  ellipse,  ES  and  E'S  will  not  in  general 
be  equal  to  each  other.  The  value  of  each,  however,  can  be 
readily  obtained  from  the  solar  tables.  The  same  tables  will 
also  give  us  the  angle  ESE',  which  is  the  angular  advance  of  the 
earth  in  its  orbit  in  the  interval.  In  the  triangle  ESE',  then, 
knowing  two  sides  and  the  included  angle,  we  can  compute  the 
sid«  EE',  and  the  angles  SEE'  and  SE'E.  The  angles  PES 
and  PE'S  having  been  obtained  by  observation,  we  can  find  the 
angles  PEE'  and  PE'E.  Then  in  the  triangle  PEE',  knowing 
two  angles  and  the  included  side,  we  can  compute  the  side  EP. 
Finally,  in  ths  triangle  PES,  knowing  an  angle  and  the  two  in- 
cluding sides,  we  can  obtain  PS,  or  the  planet's  distance  from 
the  sun,  and  the  angle  EPS,  thi  planet's  heliocentric  parallax, 
from  which,  and  the  planet's  observed  geocentric  longitude,  we 
can  obtain  the  planet's  heliocentric  longitude,  as  in  Art.  182. 
This  is,  as  we  have  already  seen,  the  heliocentric  longitude  of 
the  node. 

The  nodes  of  every  planet  are  found  to  have  a  westward 
movement,  similar  in  character  to  the  precession  of  the  equi- 
noxes. It  is  a  very  slight  movement,  however,  being  only  70'  a 
century  in  the  case  of  Mercury,  and  being  less  than  that  in  the 
case  of  the  other  planets. 

191.  Inclination  of  the  Planet's  Orbit  to  the  Ecliptic. — The 
line  of  nodes  of  any  planet  being,  as  we  have  just  now  seen,  a 
nearly  stationary  line  in  the  plane  of  the  ecliptic,  the  earth 
must  pass  it  once  in  very  nearly  six  months  in  its  revolution 
about  the  sun.  The  inclination  of  the  plane  of  any  planet's 
orbit  to  the  plane  of  the  ecliptic  may  be  determined  by  obser- 
vations made  when  the  earth  is  on  the  line  of  nodes.  In  Fig. 


PERIODIC   TIME.  163 

67  let  E  be  the  earth,  and  EN  the  line 

of  nodes  of  a  planet.     Let  AEN  be  the 

plane  of  the  ecliptic,  and  P  the  position 

of  a  planet  projected  on  the  surface  of 

the   celestial   sphere.     With   EP  as   a 

radius,  let  the  arc  PA  be  described,  per- 

pendicular  to  the  plane  of  the  ecliptic,  ris-  67- 

and  also  the  arcs  PN  and  NA.     In  the  spherical  triangle  PNA, 

right-angled  at  A,  the  arc  PA,  which  measures  the  angle  PEA, 

is  the  geocentric  latitude  of  the  planet;  AN,  or  the  angle  AEN, 

is  the  difference  between  the  geocentric  longitudes  of  the  planet 

and  the  node;  and  the  angle  PNA  is  the  inclination  of  the  plane 

of  the  orbit  to  the  plane  of  the  ecliptic.     In  the  triangle  PNA, 

we  have, 

tan  PA 


It  may  be  noticed  in  this  formula  that,  since  the  sine  of  a  small 
angle  varies  more  rapidly  than  the  sine  of  a  large  angle,  an. 
error  in  AN  will  affect  the  result  the  less,  the  greater  AN  itself 
happens  to  be  at  the  time  of  observation  :  that  is  to  say,  the 
farther  the  planet  is  from  the  node. 

192.  The  Periodic  Time  of  an  Inferior  Planet  —  The  time  in 
which  an  inferior  planet  makes  one  complete  revolution  about 
the  sun,  or  its  periodic  time,  may  be  found  by  taking  the  interval 
of  time  between  two  successive  passages  of  the  planet  through 
the  same  node.  The  accuracy  of  this  method  is  however  dimin- 
ished by  the  small  inclination  (less  than  7°)  of  the  planes  of  the 
orbits  to  the  plane  of  the  ecliptic,  which  renders  it  difficult  to 
determine  the  instant  when  the  latitude  of  the  planet  is  zero. 
It  is  found  to  be  better  to  determine  the  planet's  synodical  period, 
or  the  interval  between  two  successive  conjunctions  of  the  same 
kind,  and  from  it  to  compute  the  sidereal  period.  The  condi- 
tions of  this  problem,  and  the  method  by  which  it  is  solved,  are 
identical  with  those  in  the  case  of  the  synodical  and  the  sidereal 
period  of  the  moon,  Art.  141.  The  formula  there  given  was, 

ST 

~8+  T' 
In  applying  this  to  the  case  of  an  inferior  planet,  F  and  3 


164  MERCURY. 

denote  the  sidereal  and  the  synodical  period  of  the  planet,  and 
T  denotes,  as  before,  the  sidereal*year. 

Instead  of  using  the  interval  between  two  conjunctions  as 
the  synodical  period,  we  may  take  the  interval  between  two 
greatest  elongations  of  the  same  kind.  A  very  accurate  mean 
synodical  period  is  obtained  by  taking  two  elongations  separated 
by  a  long  interval,  and  dividing  this  interval  by  the  number 
of  synodical  periods  which  it  contains. 

193.  It  is  hardly  necessary  to  describe  the  methods  by  which 
the  other  elements  given  in  Art.  189  are  determined.     It  will 
readily  be  understood  that  the  distance  of  the  planet  from  the 
sun,  obtained  as  in  Art.  190,  or  by  Kepler's  Law,  will  enable 
us  to  obtain  the  form  and  the  magnitude  of  the  ellipse  in  which 
the  planet  moves.     The  method  by  which  the  longitude  of  the 
perihelion  is  obtained,  although  not  intrinsically  difficult,  is  too 
elaborate  for  this  work.     Finally,  when  the  longitude  of  the 
perihelion  is  obtained,  the  time  of  the  planet's  perihelion  passage 
may  evidently  at  once  be  determined.     The  perihelion  of  Venus 
has  a  very  minute  retrograde  motion :  the  perihelia  of  .all  the 
other  planets  have  an  eastward  motion,  similar  to  that  of  the 
earth's  line  of  apsides. 

MERCURY. 

194.  Mercury  revolves  about  the  sun  at  a  mean  distance  of 
about  36,000,000  miles,  the  eccentricity  of  its  orbit  being  about 
4th.     Its  synodical  period  is  about  116  days,  and  its  sidereal 
period  88  days.     Its  real  diameter  is  about  3000  miles.*     Its 
mass  is  a  matter  of  considerable  uncertainty,  and  quite  different 
values  of  it  are  given  by  different  astronomers.     It  may  be 
assumed  to  be  approximately  equal  to  y^th  of  the  mass  of  the 
earth. 

The  greatest  elongation  of  Mercury  is  only  about  28°  20' : 

*  The  distances  and  the  diameters  of  all  the  celestial  bodies  except  the  moon 
depend  for  their  accuracy  upon  the  accuracy  with  which  the  solar  parallax 
is  determined.  An  error  of  I"  in  this  parallax  would  affect  the  sun's  dis- 
tance from  us  to  the  amount  of  several  millions  of  miles,  and  propor- 
tionally also  the  distances  and  the  diameters  of  the  other  celestial  bodies ; 
the  values  given  must  therefore  be  considered  to  be  only  approximate. 


VENUS.  165 

anrl  hence  the  planet  is  rarely  visible  to  the  naked  eye,  and  is 
never  a  conspicuous  object  in  the  heavens.  It  is  not  generally 
believed  that  it  has  any  atmosphere.  It  exhibits  phases  similar 
to  those  of  the  moon,  and  due  to  the  same  causes.  It  is  asserted 
by  some  observers  that  it  rotates  on  an  axis  in  about  24  hours, 
though  the  truth  of  the  assertion  is  by  no  means  unchallenged. 
If  it  has  any  compression,  it  is  extremely  small. 

VENUS. 

195.  The  mean  distance  of  Venus   from   the   sun    is   about 
67,000,000  miles,  its  synodical  period  is  584  days,  and  its  sidereal 
period  225  days.     The  eccentricity  of  its  orbit  is  small,  being 
only  about  T¥7¥oths.     Venus  is  nearly  as  large  as  the  earth,  its 
diameter  being  7,600  miles.     Its  mass  is  about  Jths  of  the  mass 
of  the  earth.     It  has  no  perceptible  compression. 

The  greatest  elongation  of  Venus  from  the  sun  amounts  to 
about  47°  15',  and  hence  it  is  often  visible  as  an  evening  or  a 
morning  star.  At  certain  times  its  brightness  is  so  great  that 
it  can  be  seen  in  broad  daylight  with  the  naked  eye,  while  at 
night  shadows  are  cast  by  the  objects  which  it  illuminates.  It 
exhibits  phases  similar  to  those  of  Mercury. 

It  seems  to  be  generally  admitted  that  Venus  has  an  atmos- 
phere the  density  of  which  is  not  very  different  from  that  of 
the  earth's  atmosphere.  Observations  of  spots  upon  the  disc 
go  to  show  that  the  planet  rotates  upon  an  axis  in  a  period  of 
about  23^  hours;  but  this  conclusion  is  by  no  means  certain. 

The  existence  of  a  satellite  of  Venus  was  formerly  suspected, 
but  no  satellite  was  seen  at  the  transit  in  December,  1874. 

196.  Transits  of  Venus. — We   have   already  seen   (Art.   93) 
how  a  transit  of  Venus  across  the  sun's  disc  is  employed  in  de- 
termining the  distance  of  the  earth  from  the  sun.     If  the  plane 
of  the  orbit  of  Venus  coincided  with  the  plane  of  the  ecliptic, 
a  transit  would  occur  whenever  the  planet  came  into  inferior 
conjunction,  or  once  in  every  584  days.     Owing  to  the  inclina- 
tion of  the  planes  to  each  other,  however,  it  is  evident  that  at 
the  time  of  inferior  conjunction  the  planet  m^y  have  too  great  a 
latitude  to  touch  any  part  of  the  sun's  disc.    Now  the  phenomenon 
of  a  transit  of  Venus  is  analogous  to  a  solar  eclipse,  and  there- 


166  TRANSITS   OF   VENUS. 

fore  if  in  Fig.  60  we  suppose  M  to  be  Venus,  the  formula  ob- 
tained in  Art.  160  will  apply  equally  well  to  the  limits  of  the 
geocentric  latitude  of  Venus  within  which  a  transit  is  possible. 
These  limits  are  the  sum  of  the  semi-diameters  of  the  sun  and 
the  planet  and  of  the  parallax  of  the  planet,  diminished  by 
the  parallax  of  the  sun.  The  greatest  value  of  the  limits  will 
be  found  to  be  about  17'  49".  When,  therefore,  the  latitude  of 
Venus  is  more  than  17'  49"  at  the  time  of  inferior  conjunction, 
no  transit  will  occur :  and  as  Venus  in  every  sidereal  revolution 
attains  a  latitude  of  over  3°  23',  it  is  at  once  evident  that  a 
transit  is  only  a  rare  occurrence. 

197.  Intervals  between  Transits. — Since  the  latitude  of  Venus 
is  so  small  when  a  transit  occurs,  it  is  plain  that  the  planet  must 
be  either  at  or  very  near  one  of  its  nodes.  Now,  let  us  suppose 
that  Venus  is  at  its  node  at  the  time  of  inferior  conjunction, 
under  which  circumstances  a  transit  will,  of  course,  take  place. 
The  sidereal  period  of  Venus  is  224.7  days.  Now,  we  have, 

224.7d.       X  13  =  2921.11d.;  and, 

365.256d.  X    8  =  2922.05d. 

At  the  end  of  eight  years,  then,  Venus  will  be  very  near  the 
same  node  at  the  time  of  inferior  conjunction,  and  a  transit  will 
probably  occur.  Again,  we  have, 

224.7d.      X  382  =  85835.4d. 
365.256d.  X  235  =  85835.16d. ; 

so  that  transits  at  the  same  node  also  occur  every  235  years.  In 
the  same  way  transits  may  also  occur  at  the  other  node:  and  the 
intervals  between  transits  at  either  node  are  found  to  be  8,  105<}, 
8,  12H,  8,  &c.  years.*  The  longitude  of  the  ascending  node  of 
Venus  is  75°  20',  and  a  transit  at  that  node  must  occur,  when  it 
occurs  at  all,  at  the  time  when  the  earth  is  near  that  point,  which 
is  about  the  6th  of  June.  For  a  similar  reason  transits  at  the 
descending  node  occur  about  the  6th  of  December. 

The  last  two  transits  occurred  in  June,  1769,  and  December, 
1874.  The  next  two  will  occur  in  December,  1882,  and  June, 
2004.  [See  Table  VIII.,  Appendix] 

1 

*  Thus  the  years  of  transits  at  the  ascending  node  are  1761,  1769,  and 
2004:  at  the  descending  noJe,  10^9,  1874,  and  1882. 


SUPERIOR   PLANETS. 


167 


SUPERIOR   PLANETS. 

198.  The  superior  planets  are,  as  they  have  already  been  de- 
fined to  be,  planets  whose  orbits  include  the  orbit  of  the  earth. 
They  have,  like  the  inferior  planets,  superior  conjunction,  but 
can  evidently  have  no  inferior  conjunction.     Their  elongation 
from  the  sun,  eastern  and  western,  can  have  all  values  between  0° 
and  180°.     When  their  elongation  is  90°,  they  are  said  to  be  in 
quadrature,  and  when  it  is  180°,  in  opposition.    Much  that  has  al- 
ready been  said  in  this  chapter  in  reference  to  the  inferior  planets 
is  equally  true  in  reference  to  the  superior  planets.    The  elements 
of  the  orbits  are  in  both  instances  the  same,  and  so  are  the  me- 
thods by  which  the  heliocentric  longitudes  of  the  nodes  and  the 
inclinations  of  the  orbits  are  determined.     In  some  other  points 
there  is  a  difference  between  the  two  classes  of  planets;  and  these 
points  we  shall  now  proceed  to  examine. 

199.  Retrograde  Motion. — The  superior  planets,  like  the  infe- 
rior planets,  have  at  times  an  apparent  retrograde  motion,  which 

N 


Fig.  68. 


occurs  at  or  near  the  time  of  opposition.     The  explanation  of 
this  retrogradation  will  be  seen  by  a.  reference  to  Fig.  68.     S  is 


168  PERIODS   OF   A    SUPERIOR    PLANET. 

the  sun,  EE'E"E"r  the  orbit  of.  the  earth,  and  C GDP  the  orbit 
of  a  superior  planet,  the  plane  of  which  is  supposed  to  coincide 
with  the  plane  of  the  ecliptic,  and  to  meet  the  celestial  sphere  in 
the  circle  ANBM.  Let  the  earth  be  at  E,  and  the  planet  at  P, 
180°  in  geocentric  longitude  from  the  sun.  The  planet  will  ap- 
pear to  be  projected  upon  the  celestial  sphere  at  the  point  M. 
Let  both  earth  and  planet  revolve  in  their  orbits  in  the  direction 
indicated  by  the  arrow.  When  the  earth  has  reached  the  point 
Er,  the  planet,  whose  angular  velocity  is  less  than  that  of  the  earth, 
will  have  reached  some  point  P',  and  will  lie  in  the  direction  E'M': 
in  other  words,  it  has  apparently  retrograded.  If,  on  the  other 
hand,  the  earth  is  at  E"  and  the  planet  at  P,  or  in  superior  con- 
junction, the  apparent  motion  of  the  planet  is  at  that  point 
direct,  and  its  angular  velocity  appears  to  be  greater  than  it 
really  is ;  for  when  the  earth  is  at  E"r,  and  the  planet  at  P',  the 
latter,  having  in  reality  moved  through  the  arc  MM"  since  con- 
junction, appears  to  have  moved  through  the  arc  MM'". 

The  apparent  motion,  then,  of  a  superior  planet  is  direct  in 
all  cases  except  when  it  is  at  or  near  its  opposition.  The  ap- 
parent motion  of  an  inferior  planet  has  been  shown  to  be  retro- 
grade at  and  near  the  time  of  inferior  conjunction  (Art.  186). 
Now,  since  the  earth  is  a  superior  planet  to  an  inferior  planet, 
and  an  inferior  planet  to  a  superior  planet,  we  see,  by  comparing 
the  two  cases,  that  the  retrograde  motion  occurs  in  each  class 
of  planets  at  and  near  the  time  when  the  inferior  planet  conies 
between  the  sun  and  the  superior  planet. 

The  stationary  points  in  the  orbit  of  a  superior  planet  are 
identical  in  character  with  those  in  the  orbit  of  an  inferior 
planet,  and  occur  when  the  retrograde  motion  is  changing  to  the 
direct  motion,  or  the  direct  to  the  retrograde. 

200.  Synodical  and  Sidereal  Periods. — The  synodical  period 
of  a  superior  planet  is  the  interval  of  time  between  two  suc- 
cessive conjunctions  or  two  successive  oppositions.  When  a  planet 
is  in  conjunction  with  the  sun,  it  is  above  the  horizon  only  in 
the  day  time;  but  when  it  is  in  opposition,  it  is  above  the  horizon 
during  the  night,  and  can  therefore  be  readily  observed.  Hence 
in  obtaining  the  synodical  period  it  is  better  to  employ  the  in- 
terval of  time  between  two  oppositions:  and  by  determining  the 


DISTANCE    FROM    THE   SUN.  1H!) 

times  of  two  oppositions  which  are  not  consecutive,  and  div^ling 
the  interval  between  them  by  the  number  of  synodical  revo- 
lutions which  it  contains,  we  may  obtain  a  mean  value  of  tK*? 
synodical  period.  By  using  times  of  opposition  which  were  ob- 
served and  recorded  before  the  Christian  era,  a  very  accurate- 
value  of  the  mean  synodical  period  may  be  obtained. 

The  method  of  deducing  the  periodic  time  from  the  synodica-' 
period  is  the  same  that  was  used  in  the  case  of  the  inferio; 
planets  (Art.  192),  with  the  important  exception  that  in  the 
present  case  it  is  the  earth  that  gains  360°  upon  the  planet  in  the 
course  of  a  synodical  revolution,  and  not  the  planet  that  gains  it 
upon  the  earth.     If,  therefore,  we  denote  the  periodic  times  of 
the  earth  and  the  planet  by  Taud  P,  and  the  synodical  period  of 
the  planet  by  S,  we  shall  have  (Art.  141), 
360°  _  360°  _  36<T 
T         JP        "IT 


8—  T  ' 

which  gives  the  value  of  any  planet's  sidereal  period  in  terms  of 
its  synodical  period  and  the  sidereal  year. 

201.  Distance  of  a  Superior  Planet  from  the  Sun.  —  The  distance 
of  a  superior  planet  from  the  sun  may  be  obtained  by  the  method 
of  Art.  190:  or  it  may  be  obtained  from  observations  made  at 
the  time  it  is  in  opposition.     In 
Fig.  69,  let  S  be  the  sun,  E  the 


earth,  and  P  a  superior  planet 
at  the  time  of  opposition,  the 
planes  of  the  two  orbits  being  p 

supposed  to  coincide.      At  the 

end  of  a  short  interval,  let  the  earth  have  moved  to  Ef,  and  the 
planet  to  P';  the  angle  E'  OE  will  be  the  amount  of  the  apparent 
retrogradation  of  the  planet  in  that  time.  The  periods  of  the 
earth  and  the  planet  being  known,  we  can  compute  the  angular 
advance  of  each  planet  in  the  given  interval,  thus  obtaining  the 
angles  E'SE  and  P'SP.  The  radius  vector  of  the  earth's  orbit, 
SE',  can  be  found  from  the  Nautical  Almanac.  Then,  in  the 
triangle  E'SP',  we  know  the  side  E'S,  the  angle  E'SP',  which  is 
the  angular  gain  of  the  earth  on  the  planet,  and  the  angle  E'P'S, 


170  MARS. 

which  is  the  sum  of  P'SO,  or  tjie  advance  of  the  planet,  ami 
P'  OS,  or  the  apparent  retrogradation  of  the  planet.  We  can 
therefore  compute  the  side  SP',  which  is  the  distance  required. 

If  we  suppose  the  real  distance  of  the  sun  from  the  earth  not 
to  be  known,  this  method  will  still  give  us  the  ratio  between 
this  distance  and  the  distance  of  the  planet  from  the  sun.  And 
furthermore,  if  we  can  determine  the  real  distance  of  the  planet 
from  the  earth  by  observations  of  its  parallax  at  the  time  of 
opposition  (when  it  is  nearest  to  the  earth),  by  obtaining  its 
displacement  in  right  ascension  when  far  east  and  far  west  of  the 
same  meridian,  we  can  readily  obtain  the  real  distance  of  the 
earth  from  the  sun.  Such  observations  have  been  made  upon 
the  planet  Mars,  and  the  distance  of  the  earth  from  the  sun 
has  been  deduced,  as  already  given  (Art.  94). 

202.  Evening  and  Morning  Stars. — The  angular  velocity  of  a 
superior  planet  towards  the  east  is  less  than  that  of  the  earth, 
and  consequently  also  less   than   the   sun's   apparent  angular 
velocity  in  the  same  direction.     After  conjunction,  therefore,  the 
planet  will  lie  to  the  west  of  the  sun,  and  its  elongation  will 
continually  increase.     When  this  elongation  exceeds  about  30°, 
the  planet  will  begin  to  be  visible  as  a  morning  star,  and  will 
so  continue  until  it  has  fallen  180°  to  the  west  of  the  sun,  and 
is  in  opposition,     It  will  then  rise  about  sunset  and  set  about 
sunrise.     After  the  time  of  opposition  it  will  lie  more  than  180° 
to  the  west  of  the  sun,  or,  what  is  the  same  thing,  less  than  180° 
to  the  east  of  it,  and  will  rise  before  sunset.     It  will  therefore 

jvenjng  star  from  opposition  to  conjunction. 

MARS. 

203.  The  synodical  period  of  Mars  is  780  days,  and  its  side- 
real   period    687    days.      Its  mean    distance   from   the   sun   is 
141,000,000  miles,  and  the  eccentricity  of  its  orbit  about  -Jyth. 
Its  diameter  is  4,100  miles.     Different  values  are  given  to  its 
compression,  varying  between  the  limits  of  -g^th  and  ^th.     Its 
mass  is  about  Jth  of  that  of  the  earth.     It  has  two  very  small 
satellites,  discovered  by  Prof.  A.  Hall,  U.  S.  Navy,  in  1877. 

204.  Phases. — At  opposition  and  conjunction  the  same  hemi- 
sphere is  turned  towards  both  the  sun  and  the  earth,  and  con- 


MINOR   PLANETS.  171 

sequently  the  planet  appears  full.  At  quadrature  it  appears 
slightly  gibbous.  It  is  the  only  one  of  the  superior  planets 
which  exhibits  any  sensible  phases,  excepting  possibly  Jupiter. 

Mars  shines  with  a  red  light,  and  at  opposition  is  a  very  con- 
spicuous object,  sometimes  equalling  Jupiter  in  brilliancy. 

205.  Rotation,  &c. — When  examined  in  a  telescope,  the  sur- 
face of  Mars  is  seen  to  be  covered  with  patches  of  a  dull  red- 
dish color,  which  are  supposed  to  be  land,  interspersed  with 
spots  of  a  bluish  or  greenish  hue,  which  are  supposed  to  be 
water.     By  observation  of  these  spots  Mars  is  found  to  rotate 
upon  an  axis  once  in  about  24*  hours.     The  axis  of  rotation 
is  inclined  at  an  angle  of  61°  18'  to  the  plane  of  the  ecliptic, 
and  hence  there  must  be  a  change  of  seasons  not  very  different 
from  the  change  which  takes  place  on  the  earth.     White  spots 
are  seen  near  the  poles,  which  decrease  in  the  Martial  summer 
and  increase  in  the  winter.    These  spots  are  supposed  to  be  snow. 
The  existence  of  an  atmosphere  of  a  moderate  density  is  gene- 
rally admitted  by  astronomers. 

THE    MINOR    PLANETS. 

206.  Bode's  Law.— In  1778  the  astronomer  Bode,  of  Berlin, 
announced  (though  he  did  not  discover)  the  following  curious 
relation  between  the  distances  of  the  different  planets  from  the 
sun.     The  statement  of  this  relation  usually  goes  by  the  name 
of  "Bode's  law."     If  we  take  th'e  series  of  numbers 

0,  3,  6,  12,  24,  48,  96,  192,  384, 

each  of  which,  except  the  second,  is  double  the  preceding  one, 
and  add  4  to  each  of  these  numbers,  the  resulting  series, 

4,  7,  10,  16,  28,  52,  100,  196,  388, 

will  approximately  represent  the  relative  distances  of  the  planets 
from  the  sun.*  Thus  Mercury  is  36,000,000  miles  from  the 
tiun,  and  Venus  67,000,000  miles;  and  these  distances  are  to  each 
other  nearly  in  the  ratio  of  4  to  7.  There  was  however  (the 


*  The  last  two  numbers  were  not  in  the  series  as  originally  announced 
by  Bode,  since  Uranus  and  Neptune  had  not  then  been  discovered.  The 
real  distance  of  Neptune  is  one-fourth  less  than  it  should  be,  if  this  law 
were  anything  more  than  a  coincidence. 


172  MINOR   PLANETS. 

minor  plants  Demg  tken  undiscovered)  a  break  in  the  series, 
there  being  no  planet  corresponding  to  the  number  28;  and 
Bode  ventured  to  predict  that  another  planet  might  be  found  to 
exist  at  that  point  of  the  series:  that  is  to  say,  between  Mars  and 
Jupiter.  A  similar  prediction  was  made  by  Kepler,  about  the 
beginning  of  the  seventeenth  century. 

207.  Discovery  of  the  Minor  Planets. — In  1800,  six  European 
astronomers  formed  an  association  for  the  express  purpose  of 
searching  the  heavens  for  new  planets;  and  within  the  next  six 
years  four  minor  planets  were  discovered.     These  were  named 
Ceres,  Pallas,  Juno,  and  Vesta.     No  more  were  discovered  until 
the  end  of  1845,  but  since  that  time  some  have  been  discovered 
in  nearly  every  year.     The  number  discovered  up  to  October, 
1878  (including  Ceres,  &c.),  was  192. 

The  mean  distances  of  these  bodies  from  the  sun  vary  from 
200,000,000  to  300,000,000  miles.  They  are  all  very  small, 
the  largest  being  probably  not  over  300  miles  in  diameter,  and 
many  of  the  others  being  too  small  to  admit  of  measurement. 
Vesta  is  the  only  one  which  is  ever  visible  to  the  naked  eye,  and 
its  visibility  is  very  rare.  Some  of  the  others  are  so  small  that 
they  can  scarcely  be  seen  with  the  strongest  telescope,  even  at 
opposition.  Their  total  mass  is  about  one-third  of  the  earth's. 

The  French  astronomer  Leverrier  has  concluded  that  the 
mass  of  these  minor  planets  is  by  no  means  sufficient  to  produce 
the  perturbations  in  the  orbit  of  Mars  and  in  that  of  Jupiter 
which  are  believed  to  be  due  to  the  attractions  of  this  group. 
It  is  therefore  extremely  probable  that  many  other  planets, 
hitherto  undiscovered,  belong  to  the  same  cluster. 

208.  Olbers's  Theory. — Shortly  after  the  discovery  of  the  first 
four  minor  planets,  Dr.  Olbers  advanced  the  theory  that  these 
planets  were  fragments  of  a  single  planet,  which  had  been  broken 
in  pieces  by  volcanic  action  or  by  some  other  internal  force.    This 
theory  is  still  in  favor  with  many  astronomers ;  others,  however, 
object  to  it  on  the  ground  that  if  these  bodies  did  formerly  con- 
stitute one  single  body,  their  orbits  ought  to  have  a  common 
point  of  intersection,  which  is  very  far  from  being  the  case.  There 
are,  however,  certain  striking  resemblances  in  their  orbits,  which 
seem  to  argua  something  common  in  their  origin. 


JUPITER.  173 


JUPITER. 

209.  Jupiter  is  the  largest  planet  of  our  system.     At  times  it 
surpasses  Venus  in  brilliancy,  and  even  casts  a  shadow.     It  re- 
volves about  the  suflTat  a  mean  distance  of  481,000,000  miles. 
The  eccentricity  of  its  orbit  is  about  2'0 th.     Its  synodical  period  is 
399  days,  and  its  sidereal  period  4333  days,  or  about  11.9  years. 
Its  diameter  is  about  89,000  miles,  and  its  volume  is  about  1400 
times  that  of  the  earth.     It  rotates  on  an  axis  in  a  little  less 
than  ten  hours,  and  has  a  compression  of  -j^th.     Its  phases  are 
so  slight  as  to  be  scarcely  perceptible. 

210.  Belts. — When  examined  through  a  telescope,  the  disc  of 
Jupiter  is  seen  to  be  streaked  with  dark  belts,  lying  nearly 
parallel  to  the  plane  of  its  equator.     With  powerful  telescopes 
these  belts  are  found  to  have  a  gray  or  brown  tinge.     They  are 
sometimes  nearly  permanent  for  several  months,  and  sometimes 
they  change  their  shape  materially  in  the  course  of  a  few  minutes. 
There  are  usually  one  broad  and  several  narrower  belts  on  each 
side  of  Jupiter's  equator. 

It  is  generally  supposed  that  Jupiter  is  s.urrounded  by  a  dense 
atmosphere,  and  that  these  belts  are  fissures  in  this  atmosphere, 
through  which  the  dark  body  of  the  planet  is  seen.  The  distri- 
bution of  the  atmosphere  in  lines  so  nearly  parallel  to  the  equator 
is  supposed  to  be  due  to  currents  in  the  atmosphere,  similar  in 
character  to  our  trade-winds,  but  having  a  more  decided  easterly 
and  westerly  tendency,  from  the  more  rapid  rotation  and  the 
greater  size  of  the  planet.  A  point  on  Jupiter's  equator  rotates 
with  a  velocity  of  about  28,000  miles  an  hour,  while  a  point  on 
our  own  equator  rotates  with  a  velocity  of  only  about  24,000' 
miles  a  day. 

211.  Satellites. — Jupiter  is  attended  by  four  satellites  or  moons, 
revolving  about  it  from  west  to  east.     They  are  distinguished' 
from  each  other  by  the  numbers  1.,  II.,  III.,  and  IV.,  the  first  sat- 
ellite being  the  nearest  to  Jupiter.     The  second  satellite  is  about 
as  large  as  our  moon,  and  the  others  are  somewhat  larger.    They 
are  not  usually  visible  to  the  naked  eye,  though  a  few  instances 
to  the  contrary  are  on  record.     The  distance  of  the  first  satellite 
from  Jupiter  is  270,000  miles,  and  that  of  the  fourth  is  1,200,000 


,1/S.  SATELLITES   OF   JUPITER. 

miles.      The  first  revolves  about  Jkipiter  in  a  period  of  42  hours, 
a/id  thu  fourth  in  a  period  of  16d.  17h. 

212.  Phenomena  Presented  by  the  Satellites. — The  satellites,  in 
the  course  of  their  revolution  about  their  primary,  present  four 
distinct  classes  of  phenomena,  which  are  shown  in  Fig.  70.  In 
this  figure  let  S  be  the  disc  of  the  sun,  and  EE'E"E'"  the  orbit 


Fig.  70. 


of  the  earth.  Let  J  be  Jupiter,  and  ABDG  the  orbit  of  one  of 
its  satellites.  Since  the  planes  of  all  the  orbits  very  nearly 
coincide  with  the  plane  of  the  ecliptic,  we  may  consider  ABDG 
to  lie  in  that  plane.  Suppose  the  earth  to  be  at  E,  and,  in  order 
to  simplify  the  case,  suppose  it  also  to  remain  at  that  point 
during  the  short  time  required  by  the  satellite  to  revolve  about 
Jupiter. 

An  eclipse  of  the  satellite  will  occur  when  it  passes  through 
the  arc  MN,  since  it  is  then  within  the  shadow  formed  by  lines 
drawn  tangent  to  the  disc  of  the  sun  and  that  of  Jupiter.  It  may 
readily  be  calculated  that  the  length  of  the  shadow,  from  J  to  C\ 
is  about  55,000,000  miles,  so  that  the  shadow  extends  far  beyond 
the  orbit  of  the  fourth  satellite.  In  extremely  rare  cases  this 
satellite,  owing  to  the  inclination  of  its  orbit  to  the  ecliptic,  may 
fail  to  be  eclipsed. 

An  ocGultation  of  the  satellite  will  occur  when  it  passes  through 
the  arc  AB,  since  it  is  then  within  the  cone  formed  by  lines 
drawn  from  E  tangent  to  the  disc  of  Jupiter. 

A  transit  of  the  shadow  will  occur  when  the  satellite  passes 
through  the  arc  GH,  its  shadow  being  then  cast  upon  the  disc 
of  Jupiter,  and  moving  across  it  as  a  small  round  spot. 


VELOCITY   OF   LIGHT.  175 

,  a  transit  of  the  satellite  will  occur  when  it  passes 
trough  th3  arc  KL. 

It  will  evii^ntly  depend  on  the  relative  situations  of  the  sun, 
the  eaxth,  anii  Jupiter,  whether  all  these  phenomena  will  be 
observed  or  not  When  the  earth,  for  instance,  is  at  E'  or  E'"f 
it  is  plain  that  01.  ly  an  occultation  and  a  transit  will  occur. 

The  relative  situations  of  the  satellites  to  each  other  and  to 
their  primary  are  constantly  changing.  Sometimes  all  are  on 
the  same  .side  of  the  primary :  sometimes  only  one  is  visible, 
and  sometimes,  though  very  rarely,  all  are  invisible.  The  times 
at  which  these  different  phenomena  will  occur  are  computed 
beforehand,  and  are  given  in  the  American  Ephemeris,  the  time 
used  being  that  of  the  me.ridian  of  Washington.  The  longitude 
of  any  place  can  therefore  be  obtained,  at  least  approximately, 
by  observations  of  these  phenomena. 

213.  Velocity  of  Light. — If  the  transmission  of  light  is  not  in- 
stantaneous, it  is  evident  tha,t  the  same  phenomenon,  if  observed 
both  at  E'  and  E'"  (Fig.  70),  vill  not  occur  at  the  same  absolute 
instant  of  time  at  both  places,  but  will  occur  later  at  E'"  by 
the  time  required  for  light  to  cross  the  orbit  of  the  earth,  a  dis- 
tance of  185,000,000  miles.     And   such   is   actually  the  case. 
This  peculiarity  was  first  noticed  by  Romer,  a  Danish  astrono- 
mer, in  1675,  who  found  that  the  times  at  which  the  phenomena 
occurred  were  earlier  by  about  eight  minutes  at  E' ,  and  later  by 
the  same  amount  at  E'",  than  the  times  computed  for  the  mean 
distance  of  Jupiter  from  the  sun.     The  time  required  by  light  in 
passing  from  E'  to  E'"  has  been  found  by  observation  to  be  very 
nearly  16m.  27s.:  whence  the  velocity  of  light  is  calculated  to  be 
187,000  miles  a  second,  a  result  agreeing  very  closely  with  the 
velocity  obtained  from  the  constant  of  aberration,  discussed  in 
Chapter  VIII. 

214.  Mass  of  Jupiter. — The  mass  of  Jupiter  is  much  more 
accurately  known  than  the  mass  of  any  of  the  planets  which 
have  hitherto  been  described.     The  reason  is  that  Jupiter  is 
attended  by  satellites  whose  distances  from  their  primary,  and 
whose  periods  of  revolution,  can  be  obtained  by  observation. 
We  are  thus  enabled  to  compare  directly  the  attraction  which 
Jupiter  exerts  on  one  of  its  satellites  with  the  attraction  which 


176  SATUKN. 

ihe  sun  exerts  on  Jupiter;  and  a^,  by  the  law  of  gravitation,  the 
ratio  of  these  two  attractions  is  directly  as  the  ratio  of  the  masses 
of  the  two  attracting  bodies,  and  inversely  as  the  square  of 
the  ratio  of  the  distances  through  which  these  attractions  are 
exerted,  it  is  evidently  within  our  power  to  obtain  the  ratio  of  the 
two  masses.  Since  the  attraction  of  the  sun  on  Jupiter  is  equal  to 
the  centrifugal  force  of  Jupiter  in  its  orbit,  if  we  denote  the  dis- 
tance of  Jupiter  from  the  sun  by  D  and  its  sidereal  period  by  T, 
we  have,  by  the  formula  of  Art.  69,  the  expression  for  the  sun's 

4-2Z> 
attraction  on  Jupiter  equal  to  — ;™— •     In  the  same  way,  denoting 

the  distance  of  a  satellite  from  Jupiter  by  d,  and  its  period  by  t, 

we  have  for  the  attraction  of  Jupiter  on  the  satellite,  — '—  :  so 

t 

D? 

that  the  ratio  of  the  two  attractions  is  — .    Finally,  denoting 

d  J. 

the  mass  of  the  sun  by  M,  and  that  of  Jupiter  by  ra,  we  shall 
have, 

D?        M^       tf 

df*~  m   X  Z>'; 

M     m* 


m 

By  this  formula  an  approximate  value  can  be  obtained  of  the 
mass  of  any  planet  which  is  attended  by  a  satellite. 

The  mass  of  Jupiter  is  found  to  be  ju^th  of  that  of  the  sun, 
or  about  312  times  that  of  the  earth. 

SATURN. 

215.  Saturn  is,  next  to  Jupiter,  the  largest  planet  of  our  sys- 
tem, and  may  fairly  be  considered  to  be  the  most  interesting. 
It  revolves  about  the  sun  at  a  mean  distance  of  881,000,000 
miles,  in  an  orbit  whose  eccentricity  is  about  T^tli.  Its  synodical 
period  is  378  days  and  its  sidereal  period  29.45  years.  Its 
diameter  is  72,000  miles,  and  it  has  a  compression  of  about  Jth. 
It  is  attended  by  eight  satellites,  the  planes  of  whose  orbits,  with 
one  exception,  very  nearly  coincide  with  the  plane  of  its  equator. 
Two  of  these  satellites  are  rarely  visible  in  any  but  the  strongest 


RINGS    OF   SATURN.  177 

telescopes.  Their  distances  from  the  planet  range  from  121,000 
miles  to  2,300,000  miles,  and  their  sidereal  periods  from  22  hours 
to  over  79  days.  With  one  exception,  they  appear  to  be  smaller' 
than  our  moon. 

216.  Rotation,  (Sec. — Saturn  rotates  upon  an  axis  which  is  in- 
clined at  an  angle  of  about  62°  to  the  plane  of  the  ecliptic,  in  a 
period  of  10 J  hours.     Belts  are  seen  on  the  body  of  the  planet, 
similar  to  those  of  Jupiter,  although  less  marked.     Other  indi- 
cations of  the  existence  of  an  atmosphere  have  also  been  observed. 
The  mass  of  the  planet,  determined  by  the  motions  of  its  satellites, 
is  about  3sVo^n  of  that  of  the  sun,  or  about  93  times  that  of  the 
earth. 

217.  Rings  of  Saturn. — When  observed  through  a  telescope, 
Saturn  is  seen  to  be  surrounded  by  a  marvellous  system  of  lumi- 
nous rings,  lying  one  within  another  in  the  plane  of  the  planet's 
equator,  and  very  nearly  concentric  with  the  planet.     Although 
the  planet  itself  was  known  to  the  ancients,  the  existence  of  these 
rings  was  not  suspected  until  the  seventeenth  century.    They  were 
then  supposed  to  be  two  rings,  one  within  the  other ;  but  later  obser- 
vations, with  improved  instruments,  have  made  it  almost  certain 
that  the  exterior  of  these  two  rings  is  itself  composed  of  two,  and 
it  is  probable  that  similar  subdivisions  also  exist  in  the  interior 
ring.     A  third  independent  ring,  lying  within  the  others,  was 
discovered  in   1850,  by  Professor  Bond,  at  the  Observatory  of 
Harvard  College. 

Considering  only  the  two  rings  which  were  first  discovered, 
and  neglecting  the  subdivisions  that  may  exist  in  each,  the  ap- 
proximate dimensions  of  the  rings  are  as  follows : — 

Outer  diameter  of  exterior  ring 170,000  miles. 

Breadth  of  exterior  ring 10,000  miles. 

Distance  between  the  two  rings 2,000  miles. 

Outer  diameter  of  interior  ring 146,000  miles. 

Breadth    of  interior  ring 17,000  miles. 

Distance  of  interior  ring  from  Saturn 20,000  miles. 

218.  The  thickness  of  the  rings   is   very  small.     Sir  John 
Herschel  estimated  it  at  not  more  than  250  miles,  while  Professor 
Bond  considered  it  to  be  only  about  40  miles.     The  rings  appear 
to  rotate  about  the  planet  from  west  to  east,  the  period  of  rotation 


178  RINGS   OF   SATURN. 

being  about  1(H  hours,  according  toHerschel.  Various  theories 
have  been  advanced  as  to  their  composition.  The  prevailing 
theory  is  that  they  are  a  collection  of  meteoric  bodies,  revolv- 
ing about  the  planet  precisely  as  similar  rings  or  groups  of 
meteoric  bodies  are  found  to  revolve  about  the  sun.  (Chap.  XIII.) 
A  second  theory  is  that  they  are  composed  of  nebulous  matter, 
and  still  a  third  is  that  they  are  solid.  According  to  Professor 
Bond,  however,  they  are  in  a  fluid  state;  and  there  are  many 
considerations  which  favor  this  view. 

•  The  main  cause  of  the  stability  of  the  rings  in  reference  to 
the  planet  is  undoubtedly  the  centrifugal  force  induced  by  the 
rapid  rotation  above  mentioned.  Professor  Peirce,  in  developing 
the  theory  proposed  by  Professor  Bond,  maintains  that  the  per- 
manence of  the  equilibrium  is  finally  dependent  upon  the  attrac- 
tions exerted  by  the  satellites  upon  the  rings.  Herschel  says 
that  unless  they  were  originally  adjusted  in  their  present  position 
with  the  minutest  precision,  they  must  have  been  gradually 
formed  under  the  influence  of  all  the  existing  forces. 

219.  The  rings  must  present  a  magnificent  spectacle  to  the 
inhabitants  of  the  planet.     To  an  observer  on  Saturn's  equator 
they  will  appear  as   an  arch,  passing  through  the  zenith,  and 
through  the  east  and  the  west  point  of  the  horizon.    To  such  an 
observer  only  the  edge  of  the  rings  is  visible.     As  he  moves 
away  from  the  equator,  the  altitude  of  the  rings  decreases,  and 
the  side  of  the  rings  becomes  visible,  presenting  an  appearance 
not  unlike  the  familiar  one  of  the  rainbow.     Under  the  most 
favorable  circumstances  of  position,  the  rings  will  be  projected 
against  the  sky  as  an  arch  with  the  enormous  angular  breadth 
of  about  15°,  which  is  about  30  times  the  diameter  which  the 
sun  presents  to  us. 

220.  Disappearance  of  the  Rings. — As  Saturn  revolves  about 
the  sun,  the  plane  of  its  rings  remains,  like  the  plane  of  the 
earth's  equator,  fixed   in  space,  and  intersects  the  plane  of  the 
ecliptic  in  a  line  which  is  called  the  line  of  nodes  of  the  rings. 
In  Fig.  71,  let  S  be  the  sun,  ABCD  the  orbit  of  the  earth,  and 
EHLN  the  orbit  of  Saturn.     Let  HN  be  the  line  of  nodes  of 
the  rings,  and  draw  the  lines  GO  and  KM  parallel  to  HN,  and 
tangent  to  the  earth's  orbit.    When  the  planet  is  at  H,  the  plane 


RINGS    OF    SATURN. 


179 


of  the  rings  passes  through  the  sun,  and  only  the  edge  of  the 
rings  is  illuminated.  In  such  a  case  the  rings  will  disappear,  or 
at  all  events  will  only  be  seen,  in  very  powerful  telescopes,  as 
an  exceedingly  narrow  line.  Furthermore,  if,  while  the  planet 
is  within  the  lines  GO  and  KM,  the  earth  encounters  the  plane 


of  the  rings,  they  will  again  disappear.  And  thirdly,  if,  while 
the  planet  is  within  the  same  limits,  the  plane  of  the  rings 
passes  between  the  earth  and  the  sun,  the  dark  side  of  the  rings 
will  be  turned  towards  the  earth,  and  they  will  disappear- 
When  the  planet  is  beyond  these  limits,  it  is  evident  "that  the 
rings  will  always  be  visible,  and  will  present  an  elliptical  appear- 
ance, as  represented  at  E. 

Now,  we  can  readily  compute  the  length  of  time  which  Saturn 
requires  in  passing  through  the  arc  GK.  For  in  the  triangle 
CSK,  right-angled  at  C,  we  know  the  sides  OS  and  KS,  or  the 
distance  of  the  earth  from  the  sun  and  that  of  the  planet,  and  can 
therefore  obtain  the  angle  CKS.  It  will  be  found  to  oe  about  6°  V. 
This  angle  is  equal  to  the  angle  K8H\  and  therefore  double 
this  angle,  or  12°  2',  is  the  angle  through  which  Saturn  moves 
about  the  sun  in  passing  through  the  arc  GK.  Now  we  know 
that  Saturn  makes  a  complete  revolution  about  the  sun  in 


180  URANUS. 

10,759  days;  and  therefore  \ve  may  find  by  a  simple  proportion 
the  time  which  it  requires  to  pass  through  12°  2'.  This  time 
is  found  to  be  359.6  days,  or  very  nearly  a  sidereal  year;  so  that 
the  earth  makes  very  nearly  one  complete  revolution  about  the 
sun  while  Saturn  is  passing  through  the  arc  OK. 

221.  Number  of  Disappearances.  —  Since   Saturn's   period   of 
revolution  is  29.45  years,  these  disappearances  will  occur  at  in- 
tervals of  a  little  less  than  15  years.     Since  the  time  during 
which  the  planet  remains  within  the  limits  GO  and  KM  is  only 
six  days  less  than  a  year,  and  since  the  earth  may  encounter  the 
plane  of  the  ring  at  any  point  in  its  orbit,  it  is  quite  certain 
that  one  such  meeting  will   occur,  and  under  certain  circum- 
stances  there   may  be  three.     Suppose,  for  instance,  that   the 
earth  is  at  a  when  Saturn  is  at  G,  and  that  both  bodies  move 
about  the  sun  in  the  direction  EHLN.     The  earth  will  meet 
the   plane  of  the   rings   somewhere   in   the   arc    aA,  and   the 
rings  will  disappear.     The  rings  will  continue  to  be  invisible 
for  some  time,  since   their  plane  will   lie   between   the   earth 
and  the  sun.     The  earth  will  overtake  the  plane  before  Saturn 
reaches  the  point  H,  and  after  that  time  the  rings  will  be  visible 
until  the  planet  is  at  H,  when  the  plane  passes  through  the  sun, 
and  the  rings  again  disappear.     The  earth  will  now  be  near  the 
point  b,  and  the  rings  will  continue  to  be  invisible,  since  their 
dark  side  is  turned  to  the  earth.     The  earth,  passing  through  C, 
will  again  meet  the  plane  somewhere  in  the  arc  CD,  and  after 
that  time  the  rings  will  be  visible.     No  more  disappearances 
will  occur  for  about  fifteen  years,  at  the  end  of  which  interval 
the  planet  will  pass  through  the  arc  MO. 

The  last  disappearance  took  place  in  1878;  the  next  will  take 
place  in  1892. 

URANUS. 

222.  All  the  planets  which  have   thus    far   been   described, 
except  of  course  the  minor   planets,  were   known   to   the  an- 
cients; but  the  last  two   are   among  the  comparatively  recent 
discoveries   of   astronomers.      Uranus   was   discovered   by   Sir 
William  Herschel,  in    1781,    by  pure   accident.     Herschel,  as 
well  as  other   astronomers  whose  attention  was  directed  to  it, 


URANUS.  181 

at  first  supposed  it  to  be  a  comet;  and  it  was  only  after 
several  months  of  observation  that  it  was  found  to  be  a  planet. 
Several  names  were  suggested  for  it,  but  the  name  of  Uranus 
was  finally  adopted.  The  astronomical  symbol  for  it  which 
the  English  have  adopted  is  formed  from  the  initial  letter  of 
Herschel's  name. 

Upon  searching  the  star  catalogues  and  other  astronomical 
records,  it  was  found  that  the  planet  had  been  observed  no  less 
than  twenty  times  in  the  preceding  90  years,  and  had  been  con- 
sidered to  be  a  fixed  star,  its  daily  motion  being  so  slight  as  to 
have  escaped  notice.  Indeed  its  period  is  so  great  that  even  its 
annual  change  of  position  is  only  a  few  degrees. 

223.  These  previous  records,  however,  were  of  great  assistance 
to  astronomers  in  the  determination  of  the  elements  of  the 
planet's  orbit.  The  synodical  period  of  the  planet  is  370  days, 
and  the  sidereal  period  30,687  days,  or  nearly  84  years.  Its  dis- 
tance from  the  sun  is  1,800,000,000  miles,  and  its  diameter  33,000 
miles.  It  is  barely  visible  to  the  naked  eye  at  opposition. 

It  is  attended  by  four  satellites,  which  are  only  visible  in  the 
most  powerful  telescopes;  and  by  means  of  their  movements  the 
mass  of  the  planet  is  found  to  be  about  13  times  that  of  the 
earth.  One  very  remarkable  point  about  these  satellites  is  that 
their  motion  about  their  primary  is  retrograde,  or  from  east  to  west, 
in  planes  inclined  about  79°  to  the  plane  of  the  ecliptic,  while 
the  motions  of  all  the  other  satellites  which  have  hitherto  been 
described  are  from  west  to  east,  and  in  planes  making  very  small 
angles  with  the  plane  of  the  ecliptic.  Sir  Win.  Herschel  be- 
lieved that  he  had  discovered  four  other  satellites;  but  recent 
observations,  with  powerful  telescopes,  do  not  confirm  his  belief. 

Scarcely  more  can  be  said  of  the  physical  appearance  of 
Uranus  than  that  it  is  uniformly  bright.  It  exhibits  neither 
spots  nor  belts,  and  therefore  nothing  can  be  determined  as  to 
any  axial  rotation,  which  is  a  question  of  special  interest  in  the 
case  of  this  planet,  since  one  of  the  supports  on  which  what  is 
called  "  the  nebular  hypothesis"  rests  (Art.  228)  is  the  assump- 
tion that  the  satellites  of  every  planet  revolve  about  it  in  the 
same  direction  in  which  it  rotates  on  its  axis. 
16 


182  NEPTUNE. 


NEPTUNE. 

224.  Early  in  the  present  century  the  conviction  forced  itself 
upon  the  minds  of  many  astronomers  that  there  must  exist  still 
another  planet,  exterior  to  Uranus.     The  circumstance  which 
led  to  this  conclusion  was  the  existence  of  irregularities  in  the 
orbit  of  Uranus,  over  and  above  the  irregularities  which  were 
due  to  the  attractions  exerted  by  the  planets  then  known.     The 
first  systematic  attempt  to  deduce  the  elements  of  the  orbit  of 
this  unknown  planet  from  these  irregularities  seems  to  have  been 
made  by  Mr.  Adams,  of  England,  in  1843-5.     The  position  which 
he  assigned  to  the  planet  was  in  heliocentric  longitude  329°  19', 
but  this  determination  was  not  then  made  public.     The  same 
intricate  problem  was  also  solved  by  M.  Leverrier,  of  Paris,  in 
1845-6,  and  the  longitude  which  he  obtained  was  326°.     During 
the  summer  of  1846  search  was  made  for  the  planet  in  England, 
but  without  success,  owing  to  the  want  of  a  proper  star-map. 
The  observatory  at  Berlin,  however,  was  better  supplied;  and  on 
the  night  of  September  23d,  in  compliance  with  a  request  made 
in  a  letter  received  that  day  from  Leverrier,  Dr.  Galle  at  once 
detected,  in  longitude  326°  52',  what  was  apparently  a  star  of 
the  eighth  magnitude,  though  it  was  not  laid  down  on  the  map. 
Subsequent   observations    showed  that  this  body  was  really  a 
planet,  and  it  was  agreed  to  give  it  the  name  of  Neptune. 

"Such,"  in  the  words  of  Hind,  "is  a  brief  history  of  this  most 
brilliant  discovery,  the  grandest  of  which  astronomy  can  boast, 
and  an  astonishing  proof  of  the  power  of  the  human  intellect." 

225.  The  synodical  period  of  Neptune  is  367  days,  and  its 
sidereal  period  60,127  days,  or  about  164  years.     Its  mean  dis- 
tance from  the  sun  is  2,800,000,000  miles,  and  its  diameter  37,000 
miles.     It  is  attended  by  one  satellite,  and  some  astronomers 
suspect  the  existence  of  a  second.     The  mass  of  Neptune  is  about 
seventeen  times  that  of  the  earth.     The  planet  is  not  visible  to 
the  naked  eye. 

Nothing  is  yet  determined  as  to  the  physical  appearance  or 
the  axial  rotation  of  the  planet.  A  remarkable  circumstance  in 
connection  with  the  satellite  is  that,  like  the  satellites  of  Uranus, 
it  moves  about  its  primary  from  ea*t  to  west. 


NEBULAR   HYPOTHESIS.  183 

226.  )i  may  help  us  in  our  conception  of  the  immense  distance 
*t  Neptune  Irom  us,  even  when  it  is  in  opposition,  to  consider 
thai  light,  with  its  velocity  of  186,000  miles  a  second,  requires 
four  hours  to  come  from  the  planet  to  the  earth.     If  there  are 
any  inhabitants  of  Neptune,  the  sun  will  to  them  have  an  appa- 
rent diameter  of  only  ^th  of  what  it  has  to  us,  since  the  distance 
of  Neptune  from  the  sun  is  about  thirty  times  that  of  the  earth. 
It  will  therefore  appear  to  them  only  about  as  large  as  Venus 
appears,  to  us,  under  the  most  favorable  circumstances.     Saturn, 
Jupiter,  and  Uranus  may  possibly  be  visible  to  them  as  extremely 
small  bodies,  but  it  is  very  doubtful  if  any  of  the  other  planets 
of  our  system   are  visible,  even  with  the  strongest  telescopes. 
We  shall  see  hereafter,  in  the  Chapter  on  Fixed  Stars,  that,  with 
a  base-line  of  even  185,000,000  miles  (the  diameter  of  the  earth's 
orbit),  attempts  to  determine  the  distances  of  the  stars  are,  as  a 
general  rule,  wholly  unsuccessful ;  but  it  is  very  likely  that  similar 
attempts  made  by  observers  on  Neptune,  with  a  base-line  thirty 
times  as  long  as  ours,  may  give  more  satisfactory  results. 

227.  Relative  Sizes  and  Distances  of  the  Planets. — The  relative 
distances  of  the  planets  Irom  the  sun,  their  relative  magnitudes, 
as  well  as  other  numerical  data  concerning  them,  will  be  found 
in  tables  in  the  Appendix.     In  Plate  I.  will  be  seen  a  represen- 
tation of  their  relative  magnitudes,  as  they  would  appear  to  an 
observer  stationed  at  the  sanitf  distance  from  all  of  them. 

THE   NEBULA  2,   HYPOTHESIS. 

228.  Points  of  Resemblance  in  the  Planetary  Phenomena. — The 
light  of  the  planets  and  the  satellites,  when  examined  in  the 
spectroscope,  produces  only  the  ordinary  spectrum  of  reflected 
solar  light.     While,  therefore,  the  spectral  analysis  of  the  light 
of  the  sun,  the  stars,  and  other  heavenly  bodies  which  shine  by 
their  own  light,  enables  us  to  determine  to  some  extent  the  ele- 
ments of  which  they  are  composed,  a  similar  experiment  tells 
us  nothing  of  the  constitution  ot  the  planets  or  the  satellites. 
What  we  do  know,  however,  of  their  form,  their  appearance, 
their  mass,  and  their  density,  leads  us  to  conclude  that  they  are 
bodies  not  dissimilar  to  the  earth  in  general  constitution.     There 
are,  besides,  certain  remarkable  coincidences  in  the  various  phc- 


184  NEBULAR    HYPOTHESIS. 

nomena  exhibited  by  the  sun,  the  planets,  and  the  satellites, 
which  seem  to  point  to  a  common  origin  of  the  whole  solar  sys- 
tem. The  principal  of  these  coincidences  are  the  following: — 

(1.)  All  the  planets  revolve  about  the  sun  in  the  same  direction 
in  which  the  sun  rotates  upon  its  axis:  that  is  to  say,  from  west 
to  east. 

(2.)  The  planes  of  the  planetary  orbits  nearly  coincide  with 
the  plane  of  the  sun's  equator. 

(3.)  The  satellites  of  each  planet,  as  far  as  known,  revolve 
about  their  primary  in  the  same  direction  in  which  the  primary 
rotates  upon  its  axis.  The  satellites  of  Uranus  and  Neptune 
may  or  may  not  form  an  exception  to  this  rule,  for  these  planets 
are  so  distant  that  observation  fails  to  detect  any  axial  rotation 
in  them. 

(4.)  The  planes  of  the  orbits  of  the  satellites  of  each  planet 
approximately  coincide  with  the  plane  of  that  planet's  equator. 

(5.)  Both  planets  and  satellites  revolve  in  ellipses  of  small 
eccentricity. 

229.  The  Nebular  Hypothesis. — The  idea  of  the  nebular  hypo- 
thesis seems  to  have  presented  itself  at  about  the  same  time  to 
both  Sir  William  Herschel  and  Laplace.  The  principal  points 
in  it  are  the  following.  All  the  matter  which  now  composes  the 
sun,  the  planets,  and  the  satellites  once  existed  as  a  single  nebu- 
lous mass,  extending  beyond  the  present  orbit  of  Neptune,  and 
rotating  on  an  axis  from  west  to  east.  In  the  progress  of  ages 
this  nebulous  mass  slowly  contracted  and  condensed,  from  the 
loss  of  the  heat  which  it  radiated  into  space,  and  from  the 
gravitation  of  its  particles  towards  the  centre.  As  its  dimen- 
sions became  less,  its  velocity  of  rotation  became  greater,  ac- 
cording to  the  laws  of  Mechanics:  since  any  particle. moving  in 
a  circle  of  any  radius  with  a  certain  linear  velocity  would,  as  it 
approached  the  centre,  move  in  a  smaller  circle  with  nearly  the 
same  linear  velocity,  and  would  therefore  have  a  greater  angular 
velocity.  Finally,  the  centrifugal  force  generated  by  this  increased 
velocity  at  the  surface  of  the  equator  of  the  mass  exceeded  the 
attraction  towards  the  centre,  and  a  nebulous  zone  was  detached, 
which  revolved  independently  of  the  interior  mass,  just  as  the 
rings  of  Saturn  have  been  seen  to  revolve  about  that  planet.  This 


NEBULAR   HYPOTHESIS.  185 

gone,  by  concentration  at  certain  points  within  itself,  broke  up 
into  separate  masses ;  and  these  masses,  either  from  slight  differ- 
ences of  velocity  or  from  the  preponderating  attraction  of  sotne 
fraction  larger  than  the  others,  eventually  formed  one  body,  re- 
volving about  the  central  mass.  And,  furthermore,  as  these 
separate  masses  came  together,  a  motion  of  rotation  was  com- 
municated to  the  combined  mass,  just  as  a  whirlpool  or  an  eddy  is 
formed  when  two  streams  of  water  meet;  and  this  rotating  mass, 
condensing  and  contracting  in  its  turn,  threw  off  from  itself  a 
second  zone,  which  underwent  all  the  changes  above  described. 
Thus  were  formed  a  planet  and  its  satellite,  each  revolving  about 
its  primary  in  the  direction  of  that  primary's  axial  rotation :  and 
by  a  continuation  of  the  process  the  whole  system  of  planets  and 
satellites  was  evolved. 

230.  Necessary  Conditions. — It  is  a  necessary  condition  of  the 
truth  of  this  hypothesis,  that  the  planets  shall  revolve  (as  they  do 
revolve)  about  the  sun  in  the  same  direction  in  which  it  rotates. 
It  is  also  necessary  that  each  satellite  or  system  of  satellites  shall 
revolve  about  its  primary  in  the  same  direction  in  which  that 
primary  rotates.     It  is  not,  however,  absolutely  necessary  that 
the  outer  planets  shall  rotate  in  the  same  direction  in  which  they 
revolve;  although  such  a  coincidence  might  be  expected,  since 
the  revolution  of  the  outer  particles  from  which  a  planet  was 
formed  would  be  more  rapid  than  that  of  those  which  were 
nearer  to  the  sun. 

If  we  assume  this  hypothesis  to  be  true,  the  rings  of  Saturn  are 
to  be  considered  as  rings  which  did  not  form  satellites  after  they 
were  thrown  off  from  the  planet;  while  in  the  case  of  the  minor 
planets  the  ring  broke  up  into  separate  masses,  which  have  con- 
tinued to  revolve  in  independent  orbits  about  the  sun. 

231.  Experiment  in  Support  of  the  Hypothesis. — The  possible 
truth  of  the  nebular  hypothesis  is  supported  by  an  ingenious  ex- 
periment devised  by  M.  Plateau.*     A  mass  of  olive-oil  was  im- 
mersed in  a  mixture  of  alcohol  and  water,  the  density  of  the  mix- 
ture being  made  exactly  equal  to  that  of  the  oil.     In  this  way 

*  See  Annales  de  Chimie,  vol.  xxx.  (1850).  The  experiment  is  also  de- 
Bcribed  in  Carpenter's  Mechanical  Philosophy,  &c.,  one  of  the  volume!:;  of 
Bonn's  Scientific  Library  (London). 


186  NEBULAR   HYPOTHESIS. 

the  mass  of  oil  was  practically  withdrawn  from  the  influence  oi 
gravitation.  When  made  to  rofate,  the  mass  assumed  a  sphe- 
roidal form,  and  finally,  when  the  velocity  of  rotation  was  suffi- 
ciently great,  a  ring  of  matter  was  thrown  off  in  the  equatorial 
region.  This  ring  subsequently  broke  up  into  independent 
masses,  each  of  which  assumed  a  globular  form,  rotated  on  an 
axis  of  its  own,  and  continued  to  revolve  about  the  central  mass : 
thus  presenting  precisely  the  successive  phenomena  which  are 
assumed  in  the  nebular  hypothesis  to  have  occurred  in  the  for- 
mation of  the  solar  system. 

232.  The  truth  of  the  nebular  hypothesis  is  by  no  means  uni- 
versally admitted  by  astronomers  and  other  scientific  men ;  and 
it  is  difficult  to  say  what  is  the  predominant  belief  about  it  at 
the  present  time.  The  high  scientific  reputation  of  those  who 
originated  it,  and  of  those  who  have  since  supported  it,  is  suffi- 
cient justification  for  giving  it  a  place  in  this  treatise;  but  it 
must  not  be  forgotten  that  its  truth  is  still  very  emphatically  an 
open  question,  and  that  many  great  minds  are  numbered  with 
its  opponents. 

Sir  William  Herschel  was  led  to  the  adoption  of  the  nebular 
theory  by  his  examination  of  that  class  of  celestial  bodies  called 
nebulae,  some  of  which  presented  in  his  day,  and  present  now, 
the  appearance  of  masses  of  nebulous  matter.  Recent  spectro- 
scopic  examinations  of  some  of  these  nebulae  (Art.  286)  go  to 
show  that  they  are  really  what  they  seem  to  be,  masses  of  incan- 
descent vapor;  and  this  discovery  gives  a  new  interest  to  the  ne- 
bular hypothesis.  Mr.  Lockyer,  in  his  Elementary  Lessons  in 
Astronomy,  says  that  "it  may  take  long  years  to  prove  or  dis- 
prove this  hypothesis;  but  it  is  certain  that  the  tendency  of 
recent  observations  is  to  show  its  correctness." 

Fresh  doubts  are  thrown  upon  the  truth  of  the  nebular  hy- 
pothesis by  the  discovery  of  the  satellites  of  Mars  (§203);  since 
the  angular  velocity  of  the  inner  satellite  appears  to  be  three 
times  as  great  as  the  rotation  of  the  planet. 

A  statement  of  "Kirk wood's  Law,"  which  may  have  some 
bearing  on  the.nebular  hypothesis,  will  be  found  in  the  Appen- 
dix. 


PLATE  III, 


COMETS. 


1,  BIELA'S  COMET 

2,  ENCKE'S  COMET. 


3.  GREAT  COMET  OF  1861, 

4,  DONATI'S  COMET,  1858. 


COMETS.  187 


CHAPTER  XIII. 

.V^lS   AM)    METEORIC   BODIES. 
COMETS. 

233.  Ger^tc\  Description  of  Co, nets. — A  comet  is  a  body  of 
nebulous  appeal  ance  and  irreguUi  shape,  revolving  in  an  orbit 
about  the  sun.  Comets  have  usually  been  considered  to  con- 
sist for  the  most  part  of  nebulous  matter;  but  the  theory  has 
lately  been  advanced  that  they  are  collections  of  minute  meteoric 
bodies. 

Comets  differ  widely  from  each  other  in  appearance,  and  no 
description  of  them  can  be  given  to  which  thsre  will  not  be 
many  exceptions.  Generally  speaking,  a  comet  consists  of  three 
parts:  the  nucleus,  the  coma,  und  the  tail.  The  nucleus  and  the 
coma  together  form  the  head.  The  nucleus  is  a  blight  point, 
like  a  star  or  a  planet,  which  may  be  either  a  solid  mass,  or  a  mass 
of  nebulous  matter  of  a  density  greater  than  that  of  the  rest  of 
the  comet.  The  diameter  of  the  nucleus  varies  considerably  in 
different  comets:  that  of  the  comet  of  1845  (in)*  was  about 
8000  miles,  while  that  of  the  comet  of  1806  was  only  30 
miles.  The  average  value  is  not  over  500  miles:  and  in  many 
comets  no  nucleus  whatever  is  perceptible. 

The  coma  is  a  mass  of  cloud-like  matter,  mere  or  less  nearly 
globular  in  form,  which  surrounds  the  nucleus.  The  nucleus, 
however,  as  a  general  thing,  is  not  situated  at  the  centre  of  the 
coma,  but  lies  towards  that  margin  which  is  the  nearer  to  the  sun. 
The  diameter  of  the  coma  is  different  in  different  cornets:  thai 
of  the  comet  of  1847  (y)  was  only  18,000  miles,  while  that  of  tKwi 
comet  of  1811  (i)  was  over  1,000,000  miles.  Usually,  however..  i\i 

*  The  number  (in)  means  that  this  was  the  third  comet  which  appeared 
it  the  course  of  the  year. 


188  COMETS. 

is  less  than  100,000  miles.  It  is  frequently  noticed  that  the  coma 
decreases  in  apparent  diameter  as  the  comet  approaches  the  sun, 
and  increases  as  the  comet  recedes  from  it.  On  the  supposition 
that  the  coma  consists  of  vaporous  matter,  this  phenomenon  is 
explained  by  the  assumption  that  the  intense  heat  to  which  the 
comet  is  subjected  as  it  approaches  the  sun  is  sufficient  to  rarefy 
this  vaporous  matter  to  such  an  extent  that  some  of  it  becomes 
invisible. 

The  tail  is  a  train  of  cloud-like  matter  attached  to  the  head, 
which  usually  lies  in  a  direction  nearly  opposite  to  that  in  which 
the  sun  lies  from  the  head.  The  tail  is  usually  very  small  when 
the  comet  first  appears,  and  sometimes  is  not  even  perceptible. 
As  the  comet  approaches  the  sun,  the  length  of  the  tail  increases, 
and  sometimes  becomes  enormous.  In  the  comet  of  1811  (l),  for 
instance,  the  length  of  the  tail  was  100,000,000  miles ;  and  in 
that  of  1843  (i)  it  was  200,000,000  miles. 

The  angular  length  of  the  tail  depends  not  only  on  its  abso- 
lute length,  but  also  on  its  distance  from  the  earth,  and  on  the 
direction  in  which  the  axis  of  the  tail  lies.  There  are  six  comets 
on  record  of  which  the  tails  subtended  angles  of  over  90°  ;  and 
one  of  these,  that  of  1861  (n),  had  a  tail  of  104°  in  length,  as 
observed  at  some  places. 

234.  Diversity  of  Appearance. — The  description  above  given 
may  be  considered  to  apply  to  comets  taken  as  a  class ;  but,  as 
already  remarked,  important  exceptions  are  often  noticed  in 
individual  comets.  Indeed,  it  is  hardly  possible  to  compare  any 
two  comets  without  finding  marked  points  of  difference  in  them. 
Some  comets  are  not  visible  at  all,  except  by  the  aid  of  powerful 
telescopes,  and  are  hence  called  telescopic  comets ;  while  others, 
again,  are  so  conspicuous  as  to  be  visible  to  the  naked  eye  in  full 
daylight.  Some  comets  have  more  than  one  tail;  the  comet  of 
1823,  for  instance,  had  a  tail  turned  towards  the  sun,  in  addition 
to  the  usual  one  turned  from  it.  The  comet  of  1744  is  reported 
to  have  had  six  tails,  spread  out  like  an  immense  fan,  through 
an  angle  of  117°;  but  the  truth  of  the  record  is  not  above 
suspicion. 

Not  only  do  comets  differ  thus  widely  from  each  other  in  ap- 
pearance, but  even  the  same  comet  changes  its  appearance  from 


TAIL    OF   THE   COMET.  189 

day  to  day.  Sometimes  the  nucleus  decreases  in  diameter  as  it 
approaches  the  sun :  sometimes  its  brightness  increases,  and  jets 
of  luminous  matter  are  thrown  off  from  it  in  the  direction  of  the 
sun.  The  length  of  the  tail  often  increases  with  marvellous 
rapidity ;  in  the  case  of  the  Great  Comet  of  1843  (i),  the  increase 
was  estimated  to  be  about  35,000,000  miles  a  day,  after  the 
comet  had  passed  its  perihelion.  There  are  some  instances  on 
record  of  a  comet's  having  separated  into  two  distinct  comets. 
This  is  asserted  in  the  Greek  records  of  a  comet  which  appeared 
in  370  B.C.  :  and  Biela's  comet  presents  an  indubitable  instance 
of  this  kind.  This  comet  was  observed  in  1826  and  1832,  and 
was  determined  to  be  a  comet  with  a  period  of  nearly  seven  years. 
Its  return  in  1839  was  not  observed.  It  again  appeared  in  1846, 
and  then  presented  the  extraordinary  appearance  of  two  comets, 
moving  side  by  side,  at  a  distance  apart  of  over  150,000  miles. 

235.  The  Tail. — The  general  form  of  the  tail  is  that  of  a  trun- 
cated cone,  the  larger  base  being  at  the  extremity  of  the  tail.  It 
is  noticed  that  the  tail  is  always  brighter  near  the  borders  than 
along  the  middle,  from  which  it  is  inferred  that  it  is  hollow: 
since  only  on  such  a  supposition  would  the  line  of  sight  pass 
through  more  luminous  matter  when  directed  to  the  edges  than 
when  directed  to  the  middle.  With  regard  to  the  formation  of 
the  tail,  the  most  generally  accepted  theory  seems  to  be  that  the 
matter  of  which  the  nucleus  is  composed  is  excited  and  dilated 
by  the  action  of  the  sun's  rays,  as  the  comet  approaches  the  sun, 
and  that  particles  of  vaporous  matter  are  thrown  off  from  it ;  and 
that  these  particles  are  driven  to  the  rear  by  some  repulsive  force 
exerted  by  the  sun,  and  thus  form  the  tail.  What  this  repulsive 
force  exerted  by  the  sun  is,  has  not  yet  been  determined ;  but  the 
general  situation  which  the  tail  of  a  comet  has  with  reference  to 
the  sun  seems  to  justify  the  inference  that  some  such  force  does 
exist.  Nor  has  it  yet  been  determined  what  is  the  force  which 
originally  detaches  these  vaporous  particles  from  the  nucleus:  it 
may  be  the  same  repelling  force  which  drives  them  to  the  rear,  it 
may  be  a  force  generated  in  the  nucleus  itself,  or  it  may  be  a 
combination  of  both  these  forces.  If  we  adopt  the  theory  of  *ne 
meteoric  structure  of  these  bodies,  the  tail  is  to  be  considered  as  a 
cloud  of  minute  particles  of  matter,  held  together  by  their  mu- 


190 


TAIL   OF    THE    COMET. 


Fig.  72. 


tual  attraction,  or  by  the  attraction  exerted  upon  them  by  the 
denser  mass  which  constitutes  the  head. 

236.  Curvature  of  the  Tail. — The  tail  of  a  comet  is  usually  not 
straight,  but  is  concave  towards  that  part  of  space  which  the 
comet  is  leaving.  If  we  assume  the  existence  of  a  solar  repul- 
sive force,  similar  to  that  mentioned  in  the  preceding  article, 
this  peculiarity  of  shape  may  be  thus  explained.  In  Fig.  72,  let 
8  be  the  sun,  and  GCD 
a  portion  of  the  orbit  of  a 
comet.  When  the  nucleus 
is  at  A,  let  a  particle  be 
driven  from  it  in  the  di- 
rection SA,  with  a  force 
sufficient  to  carry  it  to  L 
in  the  time  in  which  the 
nucleus  moves  from  A  to  C. 
When  the  nucleus  reaches 
C,  this  particle,  still  retain- 
ing the  motion  which  it  had  in  common  with  the  nucleus,  will  be 
found  at  some  point  M.  In  the  same  way  a  particle  driven  from 
the  nucleus  when  it  is  at  B  will  be  found  at  some  point  K,  when 
the  nucleus  reaches  C:  and,  in  general,  when  the  nucleus  is  at  C 
the  tail  will  not  lie  in  the  direction  SN,  but  in  the  direction  of 
the  curve  CKM,  as  shown  in  the  figure. 

237}  Elements  of  a  Comet's  Orbit. — A  comet  is  identified  at  its 
successive  returns,  not  by  its  appearance,  which  is  liable,  as  we 
have  already  seen,  to  serious  changes,  but  by  the  elements  of  its 
orbit.  In  consequence  of  the  comparative  ease  with  which  the 
elements  of  a  parabola  can  be  calculated,  astronomers  are  in  the 
habit  of  using  that  curve  to  represent  at  first  the  approximate 
form  of  a  comet's  orbit.  The  elements  of  a  parabolic  orbit  are 
five  in  number,  and  are  as  follows : — 

(1.)  The  inclination  of  the  orbit  to  the  plane  of  the  ecliptic: 

(2.)  The  longitude  of  the  ascending  node: 

(3.)  The  longitude  of  the  perihelion: 

(4.)  The  time  at  which  the  comet  passes  its  perihelion: 

(5.)  The  distance  of  the  comet  from  the  sun  at  perihelion. 
Tables  and  formulae   have   been   constructed   by  which   these 


NUMBER   OF   COMETS.  191 

elements  can  be  computed  from  the  results  of  three  distinct  ob- 
servations of  the  position  of  the  comet:  and  these  three  observa- 
tions may  all  be  made,  if  necessary,  within  the  space  of  48  hours. 
The  parabolic  elements  having  thus  been  obtained,  the  catalogues 
of  comets  are  searched  to  see  if  these  elements  are  similar  to 
those  recorded  of  any  previous  comet.  As  it  is  highly  impro- 
bable that  the  elements  of  any  two  Comets  will  coincide  through- 
out, the  presumption  is  a  strong  one,  if  two  comets,  visible  at 
different  times,  move  in  the  same  orbit,  that  they  are  one  and  the 
same  comet:  and  the  more  often  the  coincidence  is  repeated,  the 
more  nearly  does  the  presumption  approach  to  a  demonstration. 

238.  Number  of  Comets,  and  their  Orbits. — The  number  of 
comets  which  have  been  recorded  since  the  Christian  era  is 
over  770:  and  there  are  about  80  recorded  as  observed  before 
that  date.  Of  these  850  appearances  of  comets,  some  may  un- 
doubtedly have  been  only  reappearances  of  the  same  comet: 
and,  indeed,  in  some  cases  comets  have  been  identified  with 
other  comets  previously  observed;  but  this  can  hardly  be  the 
case  with  the  majority  of  these  bodies.  Besides  these  comets 
thus  recorded,  there  must  have  been  many  others  so  situated  as 
to  be  above  the  horizon  only  in  the  day-time:  and  such  comets 
would  become  visible  only  in  case  of  the  occurrence  of  a  total 
solar  eclipse.  A  coincidence  of  this  kind  is  recorded  by  Seneca 
as  having  occurred  62  B.  c.,  when  a  large  comet  was  seen  in  close 
proximity  to  the  sun  during  a  solar  eclipse.  The  improvement 
of  telescopes  in  recent  years  has  greatly  increased  the  number 
of  comets  which  become  visible,  and  204  were  observed  between 
the  years  1800  and  1876.  We  are  justified,  therefore,  in  conclud- 
ing that  the  comets  which  have  really  come  within  our  system 
since  the  Christian  era  are  to  be  reckoned  by  thousands.  Two 
centuries  and  more  ago,  Kepler  made  the  remarkable  statement 
that  "  there  are  more  comets  in  the  heavens  than  fishes  in  the 
ocean." 

The  orbit  in  which  a  comet  moves  may  be  either  an  ellipse,  a 
parabola,  or  an  hyperbola.  The  orbits  of  329  comets  have  been 
subjected  to  mathematical  investigations,  and  the  results  of  these 
investigations  may  be  thus  tabulated :  * 

*  The  list  of  these  comets,  and  the  facts  known  concerning  them,  are 
given  in  (r.  F.  Chamliers's  D&tct'iptilt  A^trououu/  (Oxford,  Kng.). 


192  PERIODIC   TIMES   OF   COMETS. 

Comets  with  elliptical  orbits* : 20; 

Subsequent  returns  of  these  comets 66; 

Comets  with  elliptical  orbits,  which  have  not  returned     43 ; 

Comets  with  parabolic  orbits 194; 

Comets  with  hyperbolic  orbits 6. 

A  comet  whose  orbit  is  either  a  parabola  or  an  hyperbola  will 
not  return  to  our  system ;  provided,  at  least,  that  the  attraction 
of  other  bodies  does  not  alter  the  character  of  the  orbit.  It 
must  be  noticed,  however,  that  some  of  the  orbits  which  are 
called  parabolic,  may  really  be  ellipses  of  an  eccentricity  so 
great  as  to  render  their  elements  undistinguishable  from  those 
of  parabolas.  In  whatever  conic  section  a  comet  may  move, 
the  sun  is  always  at  the  focus. 

239.  Periodic  Times. — The  sixty-three  comets  which  have  been 
found  to  move  in  elliptical  orbits  differ  widely  from  each  other 
in  the  length  of  their  periods.     Among  the  twenty  comets  whose 
returns  have  been  observed,  there  are  seven  with  short  periods, 
lying  between  three  and  fourteen  years.     There  is  no  doubt  that 
these  seven  comets  are  periodic ;  but  there  is  some  uncertainty  with 
regard  to  some  others  of  the  remaining  thirteen.    Two  elements  of 
such  uncertainty  are  the  unsatisfactory  character  of  the  records 
of  the  observations  made  in  the  earlier  ages,  and  the  length  of 
time  which  the  periods  embrace,  being  often  several  hundred 
years.     Halley's  comet,  however,  with  a  period  of  about  seventy- 
five  years,  is  unquestionably  to  be  added  to  the  list  of  periodic 
comets  about  which  there  is  no  doubt.     There  are  five  other 
comets,  with  periods  not  very  different  from  that  of  Halley's, 
which  have  been  discovered  within  the  present  century,  and 
which  have  as  yet  made  no  return.     With  regard  to  the  remain- 
ing  comets   to  which   elliptical  orbits   and   periods  have  been 
assigned,  little  more  can  be  said  than  that  these  periods  embrace 
hundreds  and  even  thousands  of  years. 

240.  Motion  of  Comets  in  their  Orbits. — The  motions  of  comets 
in  their  orbits  about  the  sun  are  not  performed  in  the  same 
direction,  the  number  of  those  whose  motion  is  retrograde  being 
about  the  same  as  the  number  of  those  whose  motion  is  direct. 
According  to  Chambers,  an  examination  of  the  motions  of  the 
various  comets  shows  "that  with   comets  revolving  in  elliptic 


MAf.:S    OF    THE    COMETS.  193 

orbits  there  is  a  strong  and  decided  tendency  to  direct  notion. 
The  same  obtains  with  the  hyperbolic  orbits :  with  the  parabolic 
orbits  there  is  a  rather  large  preponderance  the  other  way  ;  and 
taking  all  the  calculated  comets  together,  the  numbers  are  too 
nearly  equal  to  afford  any  indication  of  the  existence  of  a  general 
law  governing  the  direction  of  motion." 

The  angles  which  the  planes  of  the  orbits  make  with  the 
plane  of  the  ecliptic  have  values  ranging  from  0°  to  90° ;  but 
"  there  is  a  decided  tendency  in  the  periodic  comets  to  revolve 
in  orbits  but  little  inclined  to  the  plane  of  the  ecliptic :"  and 
if  we  take  all  the  comets  into  consideration,  "wre  find  a  decided 
disposition  in  the  orbits  to  congregate  in  and  around  a  plane 
inclined  50°  to  the  ecliptic." 

Owing  to  the  great  eccentricity  of  the  orbits,  some  of  the 
comets  approach  very  near  to  the  sun  at  the  time  of  perihe- 
lion passage:  the  comet  of  1843  (i),  for  instance,  came  within 
100,000  miles  of  the  sun's  surface.  For  the  same  reason  the 
distances  to  which  some  of  the  comets  with  elliptical  orbits  recede 
from  the  sun  are  immense ;  thus  the  comet  of  1844  (n)  receded 
to  a  distance  of  400,000,000,000  miles,  over  130  times  the  dis- 
tance of  Neptune  from  the  sun.  The  velocity  of  the  comets  at 
perihelion  is  sometimes  enormous ;  this  same  comet  of  1843  swung 
about  the  sun  through  an  arc  of  180°  in  only  two  hours,  and 
moved  with  the  velocity  of  350  miles  a  second. 

241.  Mass  and  Density  of  the  Comets. — The  minuteness  of  tli3 
mass  of  the  comets  is  proved  by  the  fact  that  they  exert  no  per- 
ceptible influence  on  the  motions  of  the  planets  or  the  satellites 
although  they  sometimes  pass  very  near  to  them.  Thus  Lexell's 
comet,  1770  (i),  in  its  advance  towards  the  sun,  became  entan- 
gled with  the  satellites  of  Jupiter,  and  remained  near  them  for 
five  months,  without  sensibly  affecting  their  motions.  The  effect 
of  Jupiter's  attraction  on  the  comet,  however,  was  very  striking. 
The  comet  had  a  period  of  about  five  years,  and  yet  it  never 
appeared  after  1770.  It  was  found,  by  computation,  that  at 
its  first  return  after  that  date  it  was  so  situated  as  not  to  become 
visible;  and  that  in  1779,  before  its  second  return,  it  came 
nearer  to  Jupiter  than  Jupiter's  fourth  satellite :  and  the  pre- 
sumption is  that  its  orbit  was  so  changed  and  enlarged  that  the 


194  LIGHT   OF   THE   COMETS. 

comet  no  longer  comes  near  enough  to  the  earth  to  become  viable. 
This  same  comet  came  within  about  1,400,000  miles  of  the 
earth  in  1770 :  near  enough,  had  its  mass  been  equal  to  that 
of  the  earth,  to  increase  the  length  of  the  year  by  nearly  three 
hours  ;  but  no  sensible  effect  was  produced. 

The  mass,  then,  of  the  comets  being  so  small,  and  their 
volume  so  large,  the  density  of  the  matter  of  which  they  are 
composed  must  be  exceedingly  rare.  Indeed,  it  must  be  vastly 
more  rare  than  that  of  the  lightest  gas  or  vapor  of  which  we 
have  any  knowledge:  for  stars  of  the  smallest  magnitude  are 
distinctly  seen,  and  usually,  too,  with  no  -perceptible  diminution 
of  brightness,  through  all  parts  of  the  comets  excepting  perhaps 
the  nucleus;  and  this  too  in  cases  where  the  volume  of  nebulous 
matter  has  a  diameter  of  50,000  or  100,000  miles. 

242.  Light  of  the  Comets.  —  The  question  whether  or  not 
comets  shine  by  their  own  light  does  not  seem  to  be  satisfac- 
torily decided.  The  existence  of  phases  would  of  course  prove 
that  they  shine  by  reflected  light ;  but  although  in  one  or  two 
cases  the  statement  has  been  made  that  phases  have  been 
detected,  the  truth  of  the  statement  has  in  no  case  been  univer- 
sally accepted.  Undoubtedly  the  distance  of  some  comets  is 
so  great  that  phases  might  exist  and  still  escape  observation, 
as  in  the  case  of  the  superior  planets ;  but,  on  the  other  hand, 
some  comets  come  so  near  to  the  earth  that  there  seems  to  be 
no  good  reason  why  phases,  if  any  exist,  should  not  be  noticed. 
Observations  upon  the  light  of  the  comets  have  been  made, 
both  with  the  polariscope  and  with  the  spectroscope.  The  ob- 
servations made  with  the  polariscope  seem  to  establish  the  fact 
that  the  comets  shine,  partly  at  all  events,  by  the  reflected 
light  of  the  sun :  as,  for  instance,  the  observations  of  Airy  and 
others  on  Donati's  comet  in  1858.  Mr.  Huggins,  of  England,  to 
whom  we  owe  so  many  interesting  discoveries  made  with  the 
Bpectroscope,  has  recently  examined  the  light  of  several  comets 
with  that  instrument.  In  Brorsen's  comet  he  found  that  the 
nucleus  and  part  of  the  coma  shone  by  their  own  light.  In 
Tempers  comet  the  nucleus  shone  by  its  own  light,  and  the 
coma  by  the  reflected  rays  of  the  sun.  In  some  comets  the 
light  from  the  nucleus  resembled  that  which  conies  from  the 


-PERIODIC    COMETS. 


195 


gaseous  nebulse.  In  one  comet,  there  was  a  remarkable  resem- 
blance in  the  spectrum  produced  by  its  light  to  that  produced 
by  carbon,  not  only  in  the  position  of  the  bands,  but  in  charac- 
ter and  relative  brightness. 

Altogether,  the  question  as  to  the  light  of  the  comets  may 
fairly  be  regarded  as  still  an  open  one,  to  be  decided,  perhaps, 
by  future  observations. 


PERIODIC   COMETS. 

243.  It  has  already  been  stated,  in  Art.  239,  that  there  are  at 
least  eight  comets  which  are  undoubtedly  periodic,  seven  of  these 
being  comets  with  short  periods,  and  the  eighth  being  Halley's 
comet.  The  following  table  contains  a  list  of  these  comets. 


NAME. 

PERIOD  IN 
YEARS. 

NUMBER  OF 
APPEARANCES 
OBSERVED. 

LAST  AP- 
PEARANCE. 

Encke, 

3.3 

22 

1878 

WinneckeorPons, 

5.5 

4 

1875 

Brorsen, 

5.6 

5 

1879 

Biela, 

6.6 

6 

1852 

D'  Arrest, 

6.6 

4 

1877 

Faye, 

7.4 

5 

1873 

Mechain  or  Tuttle, 

13.7 

3 

1871 

Hal  ley, 

76. 

5 

1835 

ENCKE  8    COMET. 

244.  On  November  2fi,  1818,  a  small  and  ill-defined  telescopic 
comet  was  discovered  in  the  constellation  Pegasus,  by  the 
astronomer  Pons,  at  Marseilles.  It  remained  visible  for  seven 
weeks,  and  many  observations  were  made  upon  it.  Professor 
Encke,  of  Berlin,  finding  that  the  elements  of  the  orbit  did  not 
agree  with  those  of  a  parabola,  determined  to  subject  them  to  a 
rigorous  investigation,  according  to  the  method  proposed  by 


196  PERIODIC    COMETS. 

Gauss.  This  investigation  shewed  that  the  orbit  was  elliptical, 
and  that  the  period  of  the  comet  was  about  3  J  years.  He  fur- 
ther identified  the  comet  with  the  comets  of  1786  (i),  1795,  and 
1805,  and  predicted  that  it  would  return  to  perihelion  on  May 
24,  1822,  after  being  retarded  about  nine  days  by  the  influence 
of  Jupiter. 

"  So  completely  were  these  calculations  fulfilled,  that  astrono- 
mers universally  attached  the  name  of  '  Encke'  to  the  comet  of 
1819,  not  only  as  an  acknowledgment  of  his  diligence  and  success 
in  the  performance  of  some  of  the  most  intricate  and  laborious 
computations  that  occur  in  practical  astronomy,  but  also  to  mark 
the  epoch  of  the  first  detection  of  a  comet  of  short  period  ; — one 
of  no  ordinary  importance  in  this  department  of  science." 

The  comet  has  since  been  observed  at  every  reappearance,  the 
appearance  in  1878  being  the  twenty-second  on  record.  In 
1835,  it  passed  so  near  to  the  planet  Mercury  as  to  show  con- 
clusively that  the  generally  received  value  of  that  planet's  mass 
must  be  far  too  great:  since  the  planet  exerted  no  perceptible 
influence  on  the  comet's  orbit. 

The  comet  is  sometimes  visible  to  the  naked  eye.  It  usually 
appears  to  have  no  tail ;  but  in  1848  it  had  two,  one  about  1°  in 
length,  turned  from  the  sun,  and  the  other  of  a  less  length  and 
turned  towards  it.  At  perihelion  the  comet  passes  within  the 
orbit  of  Mercury:  while  at  aphelion  its  distance  from  the  sun  is 
nearly  equal  to  that  of  Jupiter. 

One  very  curious  feature  in  connection  with  this  comet  is  that 
its  period  is  steadily  diminishing,  by  an  amount  of  about  2£ 
hours  in  every  revolution,  the  period  having  been  nearly  1213 
days  in  1789-92,  and  only  about  1210  days  in  1862-65.  Encke's 
own  theory  to  account  for  this  diminution  is  that  the  space 
through  which  the  comet  moves  is  filled  with  some  extremely 
rare  medium,  too  rare  to  obstruct  the  motions  of  the  planets,  but 
dense  enough  to  offer  sensible  resistance  to  the  progress  of  the 
comets.  The  effect  of  this  diminution  of  velocity  is  to  diminish 
the  comet's  centrifugal  force,  so  that  the  comet  is  drawn  nearer 
to  the  sun,  and  its  orbit  becomes  smaller.  But  as  the  orbit 
becomes  less,  the  angular  velocity  of  the  comet  is  increased,  and 
its  period  of  revolution  is  decreased. 


PERIODIC   COMETS.  197 

This  theory  of  Encke's  concerning  the  existence  of  a  resisting 
medium  is  by  no  means  universally  accepted  by  astronomers. 

WINNECKE'S  OR  PONS'S  COMET. 

245.  The  second  comet  in  the  list  was  discovered  by  Pons,  on 
June  12,  1819.     Professor  Encke  assigned  to  it  a  period  of  5£ 
years,  but  the  comet  was  not  seen  again   until  March  8,  1858, 
when  it  was  detected  by  Winnecke,  at  Bonn.     He  was  at  first 
inclined  to  consider  it  a  new  comet,  but  soon  identified  it  with 
the  one  previously  discovered  by  Pons.     Its  distance  from  the 
sun  at  perihelion  is  about  70,000,000  miles,  and  its  distance  at 
aphelion  520,000,000  miles.     It  also  appeared  in  1869  and  1875. 

BRORSEN'S  COMET. 

246.  This  comet  was  discovered  by  M.  Brorsen,  at  Kiel,  on 
February  26,  1846.     The  orbit  was  found  to  be  elliptical,  with  a 
period  of  about  51  years,  and  its  return  to  perihelion  was  fixed 
for  September,  1851 ;  but  its  position  at  that  time  was  so  un- 
favorable for  observation  that  it  was   not  detected.     It  was  seen 
at  its  next  return  to  perihelion,  on  March  29,  1857.     It  again 
escaped  detection  in  1862,  but  was  seen  in  this  country  on  May 
11,  1868.    It  was  seen  also  in  1873,  and  1879. 

Its  perihelion  distance  is  60,000,000  miles,  and  its  aphelion 
distance,  530,000,000  miles. 

BIELA'S  COMET. 

247.  This  comet  was   discovered   by  M.  Biela,  an  Austrian 
officer,  at  Josephstadt,  Bohemia,  on  February  27,  1826.     It  was 
observed  for  nearly  two  months,  and  was  identified  with  comets 
which  had  previously  been  seen  in  1772  and  1805. 

Its  next  return  to  perihelion  was  fixed  for  November  27, 1832 ; 
and  the  comet  passed  perihelion  within  twelve  hours  of  that .time. 
On  October  29,  1832,  it  passed  within  20,000  miles  of  the  earth's 
orbit :  but  the  earth  did  not  reach  that  point  of  its  orbit  until  a 
month  afterwards.  No  little  alarm  was  created,  however,  outside 
of  the  scientific  world,  when  it  became  generally  known  how 
near  to  the  earth's  orbit  the  comet  would  approach. 

At  its  return  in  1839  it  was  not  observed,  owing  to  its  close  proxi- 


198  PERIODIC   COMETS. 

mity  to  the  sun.  It  was  again  Detected  on  November  28,  1845, 
and  by  the  end  of  the  year  it  was  found  to  have  separated  into 
two  parts,  and  to  present  the  extraordinary  appearance  of  two 
comets,  moving  side  by  side,  at  a  distance  apart  of  over  150,000 
miles.  It  again  returned  in  1852,  and  presented  the  same  ap- 
pearance; but  the  distance  between  the  parts  had  increased  to 
over  one  million  of  miles.  Since  that  time  the  comet  has  never 
been  seen,  unless  perhaps  in  1872,  as  a  meteoric  shower. 

Two  theories  have  been  advanced  to  account  for  this  singular 
separation.  One  is  that  the  division  may  have  been  the  result 
of  some  internal  repulsive  force,  similar  to  that  which  forms 
the  tails  of  comets;  the  other  is  that  it  may  have  been  the  result 
of  collision  with  some  asteroid.  At  perihelion  the  comet  passes 
within  the  orbit  of  the  earth,  and  at  aphelion  it  passes  beyond 
that  of  Jupiter. 

D'ARREST'S  COMET. 

248.  This  comet  was  discovered  by  Dr.  D'Arrest,  at  Leipsic, 
on  June  27,  1851.  It  remained  in  sight  for  about  three  months, 
and  its  period  was  determined  to  be  about  6?  years.  Its  return 
in  November,  1857,  was  accordingly  predicted,  and  the  predic- 
tion was  verified;  although,  owing  to  the  comet's  great  southern 
declination,  it  was  only  observed  at  the  Cape  of  Good  Hope. 
The  unfavorable  situation  of  the  comet  in  1864  prevented  its 
being  seen;  but  it  was  seen  on  its  returns  in  1870  and  1877. 

Its  perihelion  distance  is  about  100,000,000  miles,  and  its 
aphelion  distance  more  than  500,000,000  miles. 

FAYE'S  COMET. 

249.  This  comet  was  discovered  by  M.  Faye,  at  the  Paris 
Observatory,  on  November  22, 1843.  It  had  a  bright  nucleus  and 
a  short  tail,  but  was  not  visible  to  the  naked  eye.  The  elements 
of  its  orbit  were  investigated  by  Leverrier,  who  predicted  that 
it  would  return  to  perihelion  on  April  3,  1851 ;  and  it  returned 
within  about  a  day  of  the  time  predicted.  It  has  since  made 
three  returns,  viz.,  in  1858,  1865,  and  1873.  The  dimensions  of 
its  orbit  are  nearly  the  same  as  those  of  D' Arrest's  comet. 


PERIODIC   COMETS. 


199 


ME*CHAIN'S  OR  TT  ITLE'S  COMET. 

250.  This  comet  was  discovered  by  Mechain,  at  Paris,   on 
January  9,  1790.     Its  period  was  calculated  to  be  less  than  14 
years;  but  the  comet  was  not  seen  again  until  January  4,  1858, 
when  it  was  detected  by  Mr.  H.  P.  Tuttle,  at  the  Harvard  Col- 
lege Observatory.    Its  third  appearance  was  in  1871. 

H  ALLEY'S  COMET. 

251.  In  the  latter  part  of  the  seventeenth  century,  Sir  Isaac 
Newton   published  his  Principia.     In  that  great  work  he  as- 
sumed that  the  comets  were  analogous  to  the  planets  in  their 
revolutions  about  the  sun,  although  no  periodic  comet  had  then 
been  discovered.     He  explained  the  methods  of  investigating 
the  orbits  of  the  comets,  and  invited  astronomers  to  apply  these 
methods  to  the  various  comets  which  had  been  observed.     Hal- 
ley,  a  young  English  astronomer,  and  afterwards  the  second 
Astronomer  Royal,  after  a  careful  investigation,  identified  the 
comet  of  1682  with  comets  which  had  appeared  in  1531  and 
1607:  the  period   of  the  comet  being  about  75  £  years.     The 
fact  that  the  interval  of  time  between  the  first  and  the  second 
of  these  appearances  was  not  exactly  equal  to  that  between  the 
second  and  the  third  seemed  at  first  to  offer  some  difficulty; 
but  Halley,  "with  a  degree  of  sagacity  which,  considering  the 
state  of  knowledge  at  the  time,  cannot  fail  to  excite  unqualified 
admiration,"  advanced  the  theory  that  the  attractions  of  the 
planets  would  exert  some  influence  on  the  orbits  of  the  comets. 
Having  thus  decided  that  this  comet  was  a  periodic  comet,  Hal- 
ley  predicted  the  return  of  the  comet  about  the  beginning  of 
the  year  1759 ;  and  the  comet  passed  its  perihelion  on  March  12, 
in    that  year.      The    comet   again   appeared  in  1835,  and  its 
next  appearance  will  be  in  1912. 

The  comet  is  a  very  conspicuous  one,  with  a  tail  sometimes 
30°  in  length  and  sometimes  50°.  The  comet  has  been  traced 
back  through  the  astronomical  records,  with  more  or  less  cer- 
tainty, to  11  B.C.,  the  number  of  appearances  being  about  seven- 
teen. It  is  not  impossible  that  it  was  this  comet  which  appeared 
in  1066,  when  it  is  recorded  that  a  large  comet  excited  dread 


200  REMARKABLE   COMETS. 

throughout  Europe,  and  was  in  England  considered  to  presage 
the  success  of  the  Norman  invasion.  It  is  also  probably  iden- 
tical with  the  comet  )f  1456,  which  had  a  splendid  tail  60°  in 
length. 

Halley's  comet  at  perihelion  is  nearer  to  the  sun  than  Venus, 
while  at  aphelion  it  recedes  beyond  the  orbit  of  Neptune. 

REMARKABLE   COMETS   OF   THE   PRESENT   CENTURY. 
THE   GREAT   COMET  OF    1811. 

252.  The  comet  of  1811  (i)  was  discovered  on  March  26, 1811, 
and  was  visible  about  seventeen  months.     It  was  very  conspi- 
cuous in  the  autumn  of  1811,  remaining  visible  throughout  the 
night  for  several  weeks.     Sir  William  Herschel  states  that  the 
nucleus  was  well  defined,  with  a  diameter  of  about  428  miles; 
that  it  was  of  a  ruddy  hue,  while  the  surrounding  nebulous  matter 
had  a  bluish-green   tinge.     The  tail  was  about  25°  in  length, 
and  6°  in  breadth.     Its  aphelion  distance  from  the  sun  is  14 
times  that  of  Neptune,  and  its  period,  according  to  Argelander, 
is  3065  years,  with  an  uncertainty  of  43  years. 

THE   GREAT   COMET   OF    1843. 

253.  The  comet  of  1843  (i)  was  first  seen  in  the  southern 
hemisphere  in  February,  and  became  visible  in  the  northern 
hemisphere  the  next  month.     It  was  decidedly  the  most  won- 
derful comet  of  the  present  century.     Its  nucleus  and  coma 
shone  with  great  splendor,  and  its  tail  was  a  luminous  train  of 
about  60°  in  length.     On  the  day  after  its  perihelion  passage, 
and  when  only  4°  distant  from  the  sun,  it  was  seen  in  broad 
daylight  in  some  parts  of  New  England,  and  its  distance  from 
the  sun  was  measured  with  a  sextant.     It  is  described  as  having 
been  at  that  time  as  well  defined,  in  both  nucleus  and  tail,  as  the 
moon  is  on  a  clear  day.     The  comet  is  remarkable  for  its  small 
perihelion  distance,  which  was  only  about  540,000  miles ;  so  that 
the  comet  came  within  100,000  miles  of  the  sun's  surface.     The 
intensity  of  the  heat  to  which  the  comet  must  then  have  been 
subjected  is  almost  inconceivable.    Since  540,000  miles  is  about 
i-70th  of  the  distance  of  the  earth  from  the  sun,  and  the  inten- 


REMARKABLE   COMETS.  201 

sity  of  heat  varies  inversely  as  the  square  of  the  distance,  the 
heat  to  which  the  comet  was  subjected  must  have  been  about 
29,000  times  as  intense  as  the  heat  which  prevails  at  the  earth's 
surface:  a  heat  nearly  twenty  times  that  required,  as  shown  by 
experiments  with  powerful  lenses,  to  melt  agate  and  carneliau. 
For  some  days  aftei  this,  the  tail  had  a  fiery  red  appearance;  and 
its  enormous  length  of  over  200,000,000  miles,  and  the  mar- 
vellous rapidity  with  which  it  was  formed,  were  undoubtedly  the 
results  of  the  heat  which  it  endured. 

The  rapidity  with  which  this  comet  moved  about  the  sun  has 
already  been  noticed  (Arc.  240).  The  period  has  been  computed 
to  be  about  175  years. 

DGtfATl's    COMET. 

254.  This  comet,  1858  (vi),  was  discovered  on  June  2d,  by  Dr. 
Donati,  at  Florence.  It  was  then  only  discernible  with  a  tele- 
scope, but  became  visible  to  the  naked  eye  about  the  last  of 
August.  Indications  of  a  tail  began  to  be  noticed  about  the  20th 
of  August,  and  in  a  few  weeks  the  tail  assumed  a  noticeable 
curvature,  which  subsequently  became  one  of  the  most  interesting 
points  connected  with  the  comet.  The  comet  passed  its  perihelion 
on  September  29th,  and  was  at  its  least  distance  from  the  earth 
on  October  10th.  Its  tail  subtended  an  angle  of  60°,  and 
had  an  absolute  length  of  51,000,000  miles.  It  disappeared 
from  view  in  the  northern  hemisphere  in  October,  but  was  seen 
in  the  southern  hemisphere  until  March,  1859. 

This  comet  was  not  as  large  as  some  others  of  the  comets,  but 
it  was  particularly  noted  for  the  intense  brilliancy  of  its  nucleus. 
The  nebulosity  surrounding  the  nucleus  was  also  peculiar  in  its 
appearance.  It  consisted  of  seven  luminous  envelopes,  parabolic 
in  form,  and  separated  from  each  other  by  spaces  comparatively 
dark.  These  envelopes  were  detached  in  succession  from  the  comet's 
nucleus,  at  intervals  of  from  four  to  seven  days.  They  receded 
from  the  nucleus  with  the  daily  rate  of  about  1000  miles.  Per- 
fectly straight  rays  of  light,  or  "secondary  tails,"  were  also  seen. 

The  comet  has  a  period  of  about  2000  years.  A  magnificent 
memoir  of  this  comet,  by  Professor  G.  P.  Bond,  is  contained  in 
the  second  volume  of  the  Annals  of  the  Harvard  Observatory. 


202  METEORIC   BODIES. 

TIIE   GREAT   CO*MET   OF    1861. 

255.  This  comei;,  the  second  of  the  year,  was  discovered  in  the 
southern  hemisphere  on  May  13th,  but  was  not  seen  in  England 
until  June  29th,  about  two  weeks  after  its  perihelion  passage. 
The  nucleus  was  round  and  unusually  bright,  and  the  tail  at  one 
time  attained  the  length  of  over  100°.     The  comet  remained  in 
sight  for  about  a  year. 

"In  a  letter  published  at  the  time  in  one  of  the  London  papers, 
Mr.  Hind,  an  English  astronomer,  stated  that  he  thought  it  not 
only  possible,  but  even  probable,  that  in  the  course  of  Sunday, 
June  30th,  the  earth  passed  through  the  tail  of  the  comet,  at  a 
distance  of  perhaps  two-thirds  of  its  length  from  the  nucleus." 
Mr.  Hind  also  stated  that  on  Sunday  evening  there  was  noticed, 
by  both  himself  and  others,  a  peculiar  illumination  in  the  sky, 
like  an  auroral  glare;  and  a  similar  phenomenon  seems  to  have 
been  noticed  outside  of  London. 

According  to  the  observations  of  Father  Secchi,  the  light  of 
the  tail,  and  that  of  the  rays  near  the  nucleus,  presented  evidences 
of  polarization,  while  the  nucleus  itself  at  first  presented  no 
evidences  whatever ;  afterwards,  however,  the  nucleus  presented 
decided  indications  of  polarization.  Secchi  states  that  he  thinks 
this  "a  fact  of  great  importance,  as  it  seems  that  the  nucleus  on 
the  former  days  shone  by  its  own  light,  perhaps  by  reason  of  the 
incandescence  to  which  it  had  been  brought  by  its  close  proximity 
to  the  sun." 

METEORIC    BODIES. 

256.  Under  the  general  head  of  meteors  are  included  three 
classes  of  bodies : — 1st.  The  ordinary  shooting  stars,  some  of  which 
ean  be  seen  rushing  across  the  heavens  on  almost  any  clear  night ; 
2d.  Detonating  meteors,  which  are  shooting  stars,  commonly  of  an 
Anusual  size,  whose  disappearance  is  followed  by  a  sound  like  that 
•fan  explosion ;  3d.  Aerolites,  which,  after  the  flash  and  the  ex- 
plosion with  which  they  are  generally  accompanied,  are  precipi- 
tated to  the  earth  in  showers  of  stones  and   metallic  substances. 
It  is  oijsy  within  recent  years  that  the  decided  attention  of  astro- 

b.as  been  directed  to  these  bodies,  and  comparatively 


SHOOTING   STARS. 


203 


little  is  known  with  certainty  about  them  ;  but  the  ger-^ral  belief 
is  that  they  are  all  essentially  of  the  same  nature,  differing  from 
each  other  rather  in  size  and  density  than  in  other  mor^  impor- 
tant respects. 

257.  Shooting  Stars. — Scarcely  a  clear   night   passes   during 
which  shooting  stars  are  not  seen.     The  a\erage  number  of  th">?«* 
which  can  be  seen  at  any  place  by  one  observer,  on  a  cloudlet 
moonless  night,  is  estimated  to  t>e  about  six  an  hour.     There  is 
however,  an  hourly  variation  in  the  number  observed,  the  mini 
mum  occurring  about  6  P.M.,  and  the  maximum  about  6  A.M 
According  to  a  French  writer  on  this  subject,  the  mean  numbro 
of  meteors  observed  is  given  in  the  following  table : — 


HOURS  

7-8 

9-10 

11-12 

1-2 

3-4 

5-6 

P.M. 

A.M. 

MEAN    NUMBER.. 

3.5 

4 

5 

6.4 

7.8 

8.2 

.It  is  further  estimated  that  the  number  seen  at  any  one  place 
by  a  number  of  observers  sufficient  to  watch  the  whole  hemi- 
sphere of  the  heavens  is  42  an  hour,  on  the  average,  or  about 
1000  daily:  and  that  the  number  which  could  be  seen  daily  over 
the  whole  earth,  under  favorable  circumstances,  is  more  than 
8,000,000.  This  is  the  number  of  those  large  enough  to  be  vis- 
ible to  the  naked  eye :  it  is  simply  impossible  to  estimate  the 
number  of  those  -which  could  be  seen  with  the  aid  of  telescopes. 

It  is  further  noticed  that  there  are  more  shooting  stars  observed 
in  the  second  half  of  the  year  than  in  the  first.  At  certain  sea- 
sons of  the  year,  either  in  consecutive  years  or  after  the  lapse  of 
a  certain  number  of  years,  there  are  unusually  brilliant  displays 
of  these  meteors,  which  are  called  star  showers.  The  number  >f 
recognized  star  showers  now  exceeds  fifty ;  and  prominent  among 
them  are  the  shower  of  August  9-11  and  that  of  November 
11-13. 

Professor  Harkness,  of  the  Washington  Observatory,  after  an 
elaborate  investigation  of  the  quantity  of  matter  in  the  ordinary 
shooting  star,  concludes  that  it  is  not  far  from  one  grain. 

258.   The   November  Shoi'cr. — There    are    several    historical 


204  NOVEMBER   SHOWER. 

notices  of  brilliant  displays  of  meteors  which  occurred  in  the 
early  centuries  of  the  Christian  era:  and  ten  of  these,  occurring 
between  the  years  902  and  1698,  took  place  in  October  or 
November.  The  first  display,  however,  of  which  we  have  any 
detailed  account,  occurred  in  1799,  on  the  morning  of  the  13th 
of  November,  and  was  visible  over  nearly  the  whole  of  the 
western  continent.  Humboldt  witnessed  it  in  South  America, 
and  thus  describes  it : — "  Towards  the  morning  of  the  13th  we 
witnessed  a  most  extraordinary  scene  of  shooting  meteors. 
Thousands  of  bodies  and  falling  stars  succeeded  each  other 
during  four  hours.  Their  direction  was  very  regular,  from  north 
to  south.  From  the  beginning  of  the  phenomenon  there  was  not 
a  space  in  the  firmament  equal  in  extent  to  three  diameters  of 
the  moon  which  was  not  filled  every  instant  with  bodies  or  fall- 
ing stars.  All  the  meteors  left  luminous  traces  or  phosphor- 
escent bands  behind  them,  which  lasted  seven  or  eight  seconds." 

Similar  showers  also  occurred  on  the  same  day  of  the  month 
in  the  years  1831,  1832,  and  1833,  the  last  one  being  the  most 
splendid  on  record.  It  lasted  from  ten  o'clock  on  the  night  of 
the  12th  to  seven  o'clock  on  the  morning  of  the  13th,  and  was 
visible  over  nearly  the  whole  of  North  America.  The  display 
reached  its  maximum  about  four  A.M.  An  observer  at  Boston 
about  six  o'clock  counted  650  shooting  stars  in  a  quarter  of  an  hour. 
Large  fireballs  with  luminous  trains  were  also  seen,  some  of 
which  remained  visible  for  several  minutes.  Even  stationary 
masses  of  luminous  matter  are  said  to  have  been  seen :  and  one 
in  particular  is  mentioned  as  having  remained  for  some  time 
in  the  zenith  over  the  Falls  of  Niagara,  emitting  radiant  streams 
of  light. 

The  November  shower  was  witnessed  again  in  1866,  both  in 
this  country  and  in  Europe;  but  the  display  was  much  more 
brilliant  in  Europe.  The  maximum  seems  to  have  taken  place 
about  two  A.M.  on  the  14th,  when  nearly  5000  meteors  were 
counted  in  an  hour  at  Greenwich.  At  half-past  one,  124  were 
counted  in  one  minute.  The  case  was  reversed  with  the  shower 
of  1867,  the  display  being  more  brilliant  in  this  country  than  in 
Europe.  The  report  on  the  shower  from  the  United  States 
Observatory  at  Washington  states  that  as  many  as  3000  were 


HEICHT  OF    METEORS.  205 

counted  in  one  hour.  The  most  magnificent  phase  seems  to 
have  occurred  about  half-past  four  A.M.  on  the  14th.  Professor 
Loomis  states  that  at  New  Haven  about  220  a  minute  were 
counted  at  this  time.  Many  others  were  undoubtedly  rendered 
invisible  by  the  light  of  the  moon,  which  was  then  very  nearly 
full.  Most  of  the  brighter  meteors  left  trains  of  phosphorescent 
light,  which  remained  visible  for  several  seconds,  and  in  some 
cases  for  several  minutes. 

In  1868,  the  display  began  somewhat  before  midnight  on  the 
13th  and  continued  until  daybreak  on  the  14th.  Professor 
Eastman,  of  the  Washington  Observatory,  says  in  his  report, 
that  "  considering  the  number  and  brilliancy  of  the  meteors, 
their  magnificent  trains,  and  the  magnitude  of  the  meteoric 
group  through  which  the  earth  passed,  this  shower  was  unques- 
tionably the  grandest  that  has  ever  been  witnessed  at  this  Obser- 
vatory." Over  5000  meteors  were  counted,  and  it  was  estimated 
that  at  five  A.  M.  on  the  14th  the  number  falling  in  the  whole 
heavens  was  about  2500  an  hour.  Several  very  brilliant  meteors 
were  observed.  One  in  particular  was  brighter  than  Jupiter. 
It  was  at  first  of  a  deep  orange  color,  afterwards  green,  and  finally 
light  blue.  It  left  a  train  of  7°  in  length,  which  passed  through 
the  same  changes  of  color,  and  remained  visible  for  half  an 
hour.  The  paths  of  90  meteors  were  traced  upon  a  chart,  and 
were  found  in  nearly  every  instance  to  start  from  a  point  in  the 
constellation  Leo. 

259.  Height,  &c.  of  the  Meteors.  —  Concurrent  observations 
were  made  at  Washington  and  Richmond,  in  November,  1867, 
for  the  purpose  of  determining  the  parallax  of  the  meteors,  and 
thence  their  distance.  It  was  found  that  they  appeared  at  an 
average  height  of  75  miles,  and  disappeared  at  the  height  of 
55  miles.  The  velocity  with  which  they  moved  relatively  to 
the  earth  was  44  miles  a  second.  Other  observations  have  given 
nearly  the  same  results. 

The  light  of  the  meteors  is  probably  due  to  the  intense  heat 
generated  by  the  resistance  of  the  air  to  the  progress  of  these 
bodies.  Notwithstanding  the  extreme  rarity  of  the  air  at  the 
height  of  the  meteors,  it  is  still  believed  that  the  heat  resulting 

from  such  immense  velocity  is  sufficient  to  fuse  any  known  sub- 
is 


206  ORBITS    OF    METEORS. 

stance.  A  body  moving  with  tlys  velocity  at  the  earth's  surface 
would  acquire  a  temperature  of  at  least  3,000,000°.  An  exami- 
nation of  the  light  of  the  meteors  with  the  aid  of  the  spectro- 
scope, by  Mr.  A.  Herschel,  showed  that  some  of  the  meteors 
were  solid  bodies  in  a  state  of  ignition,  but  that  most  of  them 
were  gaseous. 

260.  Orbits  of  the  Meteors. — It  is  noticed  that  the  November 
meteors,  or  at  all  events  the  great  majority  of  them,  seem  to 
come  from  the  same  point  in  the  heavens, — a  point  in  the  con- 
stellation Leo.  So  also  the  August  meteors  come  from  a  point 
near  the  head  of  the  constellation  Perseus.  Such  points  are 
called  radiant  points.  Other  showers  have  also  other  radiant 
points,  situated  in  various  parts  of  the  heavens.  The  number 
of  such  points  now  recognized  is  more  than  60.  The  paths  in 
which  meteors  having  the  same  radiant  point  move  during  the 
instant  of  time  that  we  see  them,  are  really  parallel  straight 
lines,  the  apparent  convergence  of  the  paths  being  merely  the 
result  of  perspective;  in  other  words,  the  radiant  point  is  the 
vanishing  point  (Art.  16)  of  these  parallel  lines. 

Knowing  the  direction  and  the  velocity  with  respect  to  the 
earth  of  the  motion  of  a  meteor,  it  is  easy  to  compute  the  same 
elements  of  its  motion  with  reference  to  the  sun.  The  results  of 
such  computation,  together  with  the  existence  of  the  radiant 
points  and  the  periodic  recurrence  of  showers,  have  led  to  the 
theory  that  the  November  meteors  are  collected  in  a  ring,  or  in 
several  rings,  or  possibly  in  a  series  of  clusters  or  groups,  which 
revolve  about  the  sun ;  and  that  the  showers  occur  when  the 
earth  encounters  these  rings  or  groups.  The  theory  of  one  ring  is 
exemplified  in  Fig.  73.*  Let  ABCD  represent  the  orbit  of  the 
earth,  and  A  GBE  a  ring  of  meteors  revolving  about  the  sun. 
If  A  and  B  are  the  points  at  which  the  earth  enters  the  ring, 
there  will  be  displays  of  meteors,  when  the  earth  is  at  these 
points,  similar  to  the  August  and  the  November  shower.  Unless 
the  plane  of  the  ring  coincides  with  the  plane  of  the  ecliptic, 
there  will  be  no  showers  at  any  other  points  of  the  earth's  orbit. 
If  we  imagine  the  ring  to  be  broken,  or  to  be  of  unequal 

*  Phipson's  Meteors,  Aerolites,  and  Falling  Stars.     London,  1867. 


ORBITS   OF   METEORS.  207 

thickness,  or  to  consist  of  a  series  of  groups,  we  may  account  for 
the  irregularities  which  exist  in  the  annual  showers ;  and  by 


Fig.  73. 

supposing  the  ring  to  have  a  period  of  about  331  years,  we  may 
account  for  the  extraordinary  displays  of  meteors  which  happen 
in  about  that  interval  of  time.  The  duration  of  a  shower,  and  the 
known  velocity  of  the  earth  in  its  orbit,  enable  us  to  obtain  an 
approximate  value  of  the  breadth  of  the  ring.  Professor  East- 
man estimates  that  the  breadth  of  that  portion  of  the  ring  through 
which  the  earth  passed  in  November,  1868,  could  not  have 
been  less  than  115,000  miles.  The  breadth  of  the  stream  in 
1867  was  less  than  this,  but  more  densely  packed  with  meteors. 

Leverrier,  a  French  astronomer,  has  computed  the  elements 
of  the  orbit  of  the  November  meteors.  He  finds  the  major 
semi-axis  to  be  10.34,  the  perihelion  distance  0.989  (the  radius 
of  the  earth's  orbit  being  unity),  and  the  eccentricity  0.9044. 
This  would  carry  the  aphelion  beyond  the  orbit  of  Uranus,  if 
both  orbits  were  projected  upon  the  plane  of  the  ecliptic. 

According  to  Professor  Loomis,  the  relative  situations  of  the 
orbit  of  the  November  meteors  and  the  orbits  of  the  earth  and 
the  other  planets,  are  represented  in  Fig.  74 ;  the  orbit  of  the 
meteors  being  a  very  eccentric  ellipse,  the  aphelion  of  which 
lies  beyond  the  orbit  of  Uranus,  and  the  period  of  the  me- 
teors being  331  years.  According  to  the  same  authority,  the 
August  meteors  revolve  in  a  similar  but  much  more  eccentric 
ellipse,  of  which  the  aphelion  lies  far  beyond  the  orbit  of  Nep- 
tune. The  theory  has  also  been  advanced  that  meteors  (or,  at 
all  events,  some  of  them)  are  to  be  regarded  as  satellites  of 


208 


DETONATING   METEORS. 


^Orlnt  of  Uranus? 


Orbit  of  Jupifer 


the  earth  rather  than  of  the  sum.      On  this  subject  Sir  John 
Herschel  says,  in  his  Outlines  of  Astronomy : — "It  is  by  no  means 

inconceivable  that  the  earth, 
approaching  to  such  as  dif- 
fer but  little  from  it  in  direc- 
tion and  velocity,  may  have 
attached  them  to  it  as  per- 
manent satellites,  and  of  these 
there  may  be  some  so  large 
as  to  shine  by  reflected  light, 
and  to  become  visible  for  a 
brief  moment;  suffering,  after 
that,  extinction  by  plunging 
into  the  earth's  shadow." 

261.  Detonating  Meteors. — 
The  height  and  the  velocity 
of  these  bodies  are  not  essen- 
tially different  from  those  of 
the  ordinary  shooting  stars. 
They  are,  however,  generally 
of  an  unusual  brilliancy,  and 
their  appearance  is  followed 
by  an  explosion,  or  a  series 
of  explosions,  the  intensity 
of  which  is  sometimes  terrific. 
Records  of  more  than  eight 
hundred  detonating  meteors 
are  to  be  found  in  scientific  journals.  The  phenomena  connected 
with  the  appearance  of  these  bodies  are,  however,  so  nearly  iden- 
tical in  character,  that  one  instance  may  suffice  to  exemplify  all. 
"On  the  2d  of  August,  1860,  about  10  P.M.,  a  magnificent  fire- 
ball was  seen  throughout  the  whole  region  from  Pittsburg  to 
New  Orleans,  and  from  Charleston  to  St.  Louis,  an  area  of  900 
miles  in  diameter.  Several  observers  described  it  as  equal  in 
size  to  the  full  moon,  and  just  before  its  disappearance  it  broke 
into  several  fragments.  A  few  minutes  after  the  flash  of  the 
meteor  there  was  heard  throughout  several  counties  of  Ken- 
tucky and  Tennessee  a  tremendous  explosion,  like  the  sound  of 


AEK3LITES.  209 

distant  cannon.  Immediately  another  noise  was  heard,  not 
quite  so  loud,  and  the  sounds  were  re-echoed  with  the  prolonged 
roar  of  thunder.  From  a  comparison  of  a  large  number  of 
observations,  it  has  been  computed  that  this  meteor  first  became 
visible  over  Northeastern  Georgia,  about  82  miles  above  the 
earth's  surface,  and  that  it  exploded  over  the  southern  boundary 
line  of  Kentucky,  at  an' elevation  of  28  miles.  The  length  of 
its  visible  path  was  about  240  miles,  and  its  time  of  flight  eight 
seconds :  showing  a  velocity  relative  to  the  earth  of  30  miles  per 
second.  It  is  hence  computed  that  its  velocity  relative  to  the 
sun  was  24  miles  per  second." 

The  explosions  are  probably  due  to  the  sudden  compression 
and  shocks  to  which  the  air  is  subjected  as  the  meteor  rushes 
through  it,  as  happens  when  a  gun  is  fired ;  or  to  the  rushing 
of  the  air  into  the  vacuum  which  the  body  creates  in  its  rear. 
The  appearance  of  these  bodies  is  so  sudden,  and  their  velocity 
so  great,  that  it  is  almost  impossible  to  obtain  any  definite  value 
of  their  magnitude.  The  diameters  of  some  of  them  are  stated 
to  have  been  several  thousand  feet  in  length,  but  the  estimate 
must  be  taken  with  considerable  caution,  particularly  as  it  is 
impracticable  to  distinguish  between  the  meteor  itself  and  the 
blaze  of  light  which  surrounds  it. 

262.  Aerolites. — Although  the  ordinary  shooting  stars  some- 
times appear  to  break  in  pieces,  there  is  no  evidence  that  any 
part  of  them  falls  to  the  earth.  But  occasionally  solid  masses 
of  stone  or  of  metallic  substances  do  fall  to  the  earth,  their  fall 
being  usually  preceded  by  the  flash  and  the  discharge  of  a  deto- 
nating meteor.  There  is  no  doubt  whatever  about  the  authen- 
ticity of  most  of  these  cases,  and  the  record  of  them  extends 
far  back  into  ancient  history.  A  fall  of  meteoric  stones  near 
Home,  650  years  before  Christ,  is  mentioned  by  the  historian 
Livy;  and  a  large  block  of  stone  is  said  to  have  fallen  in 
Thrace,  near  what  is  now  called  the  Strait  of  Dardanelles,  465 
years  before  Christ.  The  entire  number  of  aerolites  of  which 
we  have  any  determinate  knowledge  is  more  than  400 ;  and  more 
than  twenty  falls  of  aerolites  have  occurred  in  the  United  States 
since  the  beginning  of  the  present  century.  The  British  Mu- 
seum contains  a  large  collection  of  aerolites,  one  of  which 


210  AEROLITES. 

weighs  8287  pounds ;  and  many  other  similar  specimens  are  to 
be  found  in  the  cabinets  of  colleges  and  museums,  both  in  this 
country  and  in  Europe.  The  following  are  instances  of  falls 
Which  have  occurred  since  1800. 

In  1807,  on  the  morning  of  December  14th,  a  brilliant  meteor, 
with  an  apparent  diameter  equal  to  about  one-half  of  that  of 
the  moon,  was  seen  moving  over  the 'town  of  Weston,  in  the 
southwestern  part  of  Connecticut.  After  its  disappearance, 
three  loud  explosions  were  heard,  followed  by  a  continuous 
rumbling.  Fragments  of  stone  were  precipitated  to  the  earth 
within  an  area  of  a  few  miles  in  diameter..  The  entire  weight 
of  these  fragments  is  estimated  to  be  about  300  pounds.  One 
fragment,  weighing  36  pounds,  is  preserved  in  the  museum 
at  Yale  College.  The  specific  gravity  of  the  aerolite  was  about 
3-},  and  among  its  components  were  silex,  oxide  of  iron,  magne- 
sia, nickel,  and  sulphur. 

On  the  1st  of  May,  1860,  about  noon,  there  was  a  number 
of  explosions  over  the  southeastern  part  of  Ohio.  Stones  were 
seen  to  fall  to  the  earth,  and  in  some  cases  they  penetrated  the 
earth  to  a  distance  of  three  feet.  About  thirty  fragments  were 
found,  the  largest  of  which  weighs  103  pounds,  and  is  to  be 
found  in  the  cabinet  of  Marietta  College.  The  combined  weight 
of  these  thirty  fragments  is  not  far  from  700  pounds,  and  the 
specific  gravity  and  the  composition  are  very  similar  to  those  of 
the  Weston  aerolite. 

Another  phenomenon  of  this  character  occurred  in  Piedmont, 
on  February  29,  1868,  about  the  middle  of  the  forenoon.  There 
was  a  heavy  discharge  like  that  of  artillery,  followed,  after  a 
short  interval,  by  a  second  discharge.  A  mass  of  irregular 
shape  was  seen  in  the  air,  enveloped  in  smoke,  and  followed  by 
a  long  train  of  smoke.  Other  bodies,  similar  in  appearance  to 
meteors,  were  also  seen.  The  analysis  of  the  stones  which  fell 
showed  the  existence  in  them  of  the  components  mentioned 
above,  and  also  of  copper,  manganese,  and  potassium. 

Other  aerolites  have  been  subjected  to  chemical  analysis. 
Of  the  65  elementary  substances  known,  24  at  least  have  been 
found  in  aerolites,  and  no  new  elements  have  been  discovered. 
A  meteoric  shower  usually  consists  of  meteoric  iron  and  meteoric 


CONNECTION   OF   COMETS   AND   METEORS.  211 

Ftone ;  the  iron  is  an  alloy  of  which  the  principal  part  is  nickel, 
and  which  also  contains  cobalt,  tin,  copper,  manganese,  and 
carbon ;  the  stone  contains  chiefly  those  minerals  which  are 
abundant  in  lava  and  trap-rock.  The  proportions  in  which 
these  ingredients  enter  into  the  composition  of  different  aerolites 
differ  greatly :  sometimes  an  aerolite  contains  96  per  cent,  of 
iron,  sometimes  scarcely  any  iron  at  all.  A  substance  called 
schreibersite,  which  is  a  compound  of  iron,  nickel,  and  phos- 
phorus, is  always  found  in  these  bodies. 

The  explosion  of  an  aerolite  may  be  due  either  to  the  intense 
heat  generated  by  its  rapid  motion,  or  to  the  pressure  to  which 
it  is  subjected  by  the  resistance  of  the  atmosphere. 

263.  Origin  of  Aerolites. — Many  theories  have  been  advanced 
to  explain  the  origin  of  these  bodies.     One  theory  is  that  they 
may  be  formed  in  the  atmosphere  by  the  aggregation  of  minute 
particles  drawn  up  from  the  surface  of  the  earth ;  but  one  objec- 
tion to  this  theory  is  that  it  does  not  account  for  the  nearly 
horizontal   direction  in  which   they  move,  and  for   the   great 
velocity  of  their  motion.     A  second  theory,  that  they  are  thrown 
from  terrestrial  volcanoes,  is  open   to  the  same  objection.     A 
third  theory,  that  they  may  be  ejected  from  the  volcanoes  of  the 
moon,  is  weakened  by  the  fact  that  observation  shows  no  signs 
(or  at  least  almost  no  signs)  of  activity  in  the  lunar  volcanoes. 
The  most  probable  theory  is  that  they  are,  like  the  planets  and 
the   comets,  satellites  of  the  sun,  revolving  about  it  in  orbits 
which  intersect  the  orbit  of  the  earth,  and  that  their  fall  to  the 
earth's  surface  is  either  the  direct  result  of  their  own  motion, 
or  is  due  to  the  resistance  of  the  atmosphere,  and  the  attraction 
exerted  upon  them  by  the  earth. 

264.  Possible  Connection  of  Comets  and  Meteoric  Bodies. — The 
facts  which  have  been  presented  in  this  chapter  in  relation  to 
comets  and  meteoric  bodies  point  to  one  certain  conclusion : — 
that  space,  or  at  least  that  portion  of  space  through  which  the 
earth  moves,  must  be  considered  to  be  filled  with  a  countless 
number  of  comparatively  minute  bodies,  the  aggregate  mass  of 
which  cannot  fail  to  be  very  great.     In  1848,  Dr.  Mayer,  of 
Germany,  advanced  a  theory  that  the  light  and  the  heat  of  the 
sun  are  caused  by  the  incessant  collision  of  meteoric  bodies  with 


212  CONNECTION   OF   COMETS  AND   METEORS. 

its  surface.  In  connection  with.this  subject,  Professor  William 
Thompson  states  that  if  the  earth  were  to  tall  into  the  sun,  th« 
amount  of  heat  generated  by  the  shock  would  be  equal  to  that 
which  the  sun  now  gives  out  in  95  years ;  and  that  the  planet 
Jupiter,  under  similar  circumstances,  would  generate  an  amount 
of  heat  equal  to  that  given  out  by  the  sun  in  32,000  years. 

There  is  a  striking  similarity  between  the  elements  of  the 
orbit  of  TempePs  comet  and  those  of  the  orbit  of  the  Novem- 
ber shower,  mentioned  in  Art.  260.  There  is  also  a  similar 
coincidence  in  the  orbit  of  the  August  shower  and  that  of  the 
Great  Comet  of  1862 ;  and  the  opinion  is  gaining  ground  with 
astronomers,  not  only  that  each  of  these  comets  leads  the  group 
with  the  elements  of  whose  orbit  its  own  elements  so  nearly 
coincide,  but  also  that  there  is  a  close  connection,  generally, 
between  comets  and  meteoric  bodies.  The  following  statement 
of  this  new  theory  is  taken  from  an  article  by  Professor  Simon 
Newcomb,  in  the  North  American  Review  of  July,  1868. 

"The  planetary  spaces  are  crowded  with  immense  numbers 
of  bodies  which  move  around  the  sun  in  all  kinds  of  erratic 
orbits,  and  which  are  too  minute  to  be  seen  with  the  most  power- 
ful telescopes. 

"If  one  of  these  bodies  is  so  large  and  firm  that  it  passes 
through  the  atmosphere  and  reaches  the  earth  without  being 
dissipated,  we  have  an  aerolite. 

"  If  the  body  is  so  small  or  so  fusible  as  to  be  dissipated  in 
the  upper  regions  of  the  atmosphere,  we  have  a  shooting  star. 

"  A  crowd  of  such  bodies  sufficiently  dense  to  be  seen  in  the 
sunlight  constitutes  a  comet. 

"A  group  less  dense  will  be  entirely  invisible,  unless  the 
earth  happens  to  pass  through  it,  when  we  shall  have  a  meteoric 
shower." 

It  is  not  impossible  to  conceive  that  the  planets  themselves 
may  have  been  formed  by  the  aggregation  of  these  minute 
bodies,  a  method  of  formation  exactly  the  opposite  of  that 
which  is  set  forth  in  the  nebular  hypothesis.  An  article  in 
which  such  an  origin  is  suggested  for  the  planets,  the  comets, 
and  Saturn's  rings,  will  be  found  in  the  North  American  Review 
for  1864. 


FIXED   STARS.  213 


CHAPTER  XIV. 

THE   FIXED   STARS.      NEBULA.      MOTION  OF  THE  SOLAR  SYSTEM. 

265.  We  have  now  examined  the  motions  and  the  orbits  of 
all  the  known  members  of  the  solar  system.  We  have  seen  that 
the  planets  and  their  satellites,  besides  their  apparent  diurnal 
motion  towards  the  west  in  orbits  whose  planes  are  perpen- 
dicular to  the  axis  of  the  celestial  sphere,  have  also  independent 
motions  of  their  own  in  elliptical  orbits,  the  sun  being  at  the 
common  focus  of  the  orbits  of  the  planets,  and  each  planet  being 
at  the  common  focus  of  the  orbits  of  its  satellites.  Besides  these 
bodies,  there  is  a  vast  number  of  other  bodies  visible  in  the 
heavens,  the  phenomena  presented  by  which  are  radically  differ- 
ent from  those  which  have  hitherto  been  noticed.  Continuous 
observations  have  been  made  upon  the  stars  from  year  to  year, 
and  even  from  century  to  century ;  and  it  has  been  found  that, 
after  the  results  of  these  observations  have  been  freed  from  the 
effects  of  precession  and  nutation  (which,  by  shifting  the  position 
of  the  points  or  the  planes  of  reference,  may  give  the  stars  an 
apparent  motion),  the  real  change  of  position  of  the  stars  is 
extremely  small.  Sirius,  the  brightest  star  in  the  whole  heavens, 
has  an  annual  motion  of  V ;  a  Centauri,  the  brightest  star  in 
the  southern  hemisphere,  has  an  annual  motion  of  nearly  4" ;  61 
Cygni  and  s  Indi,  both  small  stars,  have  annual  motions  of 
5"  and  1"  respectively.  In  only  about  30  stars,  however,  has 
the  amount  of  this  change  of  position  been  found  to  be  greater 
than  1"  a  year ;  and  in  the  others,  if  any  motion  at  all  is  de- 
tected, it  is  only  that  of  a  few  seconds  in  a  century.  These 
motions  are  called  proper  motions,  to  distinguish  them  from 
those  which  are  only  apparent,  and  the  stars  are  called  fixed 
stars :  a  term  which  must  be  understood  to  imply,  not  that 
they  have  no  motion  in  space,  but  that  whatever  motion  they 


214  FIXED   STARS. 

have  makes  no  perceptible  alteration  in  their  position  upon  the 
celestial  sphere. 

When  a  star  moves  obliquely  to  the  line  joining  the  earth  and 
the  star,  its  motion  in  its  orbit  can  be  resolved  into  two  motions: 
one  along  the  line  of  sight,  either  directly  towards  the  earth  or 
directly  from  it,  and  the  other  at  right  angles  to  that  line.  This 
latter  motion  will  cause  the  star  to  shift  its  apparent  position 
upon  the  celestial  sphere,  and  will  be  the  proper  motion  above 
described ;  or,  as  it  may  be  called,  the  transverse  proper  motion. 
Now,  it  is  evident  that  in  order  to  obtain  the  real  motion  of  any 
star  in  space  we  must  be  able  to  determine,  not  only  its  trans- 
verse motion,  but  also  its  motion  towards  or  from  the  earth.  As 
long  as  the  detection  of  such  a  motion  depended  upon  our  ability 
to  detect  either  an  increase  or  a  diminution  of  the  star's  bright- 
ness, its  immense  distance  from  us  rendered  the  task  a  hopeless 
one ;  but  very  recently  the  spectroscope  has  afforded  us  the  means 
of  solving  this  problem.  The  details  of  this  method  will  be 
given  at  the  end  of  this  Chapter. 

266.  The  Number  of  the  Fixed  Stars. — The  number  of  stars  in 
the  entire  sphere  which  are  visible  to  the  naked  eye  is  between 
6000  and  7000,  according  to  the  largest  estimate ;  but  the  number 
of  those  visible  at  any  one  time  at  any  place  is  less  than  3000. 
By  means  of  telescopes,  thousands  and  even  millions  of  other 
stars  are  brought  into  view,  in  such  numbers  as  almost  to  defy 
any  attempt  at  computation. 

267.  Magnitudes. — The  fixed  stars  are  classified  arbitrarily  by 
astronomers  according  to  their  relative  brightness,  the  different 
classes  receiving  the  name  of  magnitudes,  and  the  first  magnitude 
comprising  those  stars  which  are  the  brightest.     Different  astro- 
nomers, however,  sometimes  assign  different  magnitudes  to  the 
same  star.     According  to  Argelander's  classification,  there  are 
20  stars  of  the  first  magnitude,  65  of  the  second,  190  of  the  third, 
425  of  the  fourth,  &c.,  the  numbers  in  the  following  magnitudes 
increasing  very  rapidly.     It  is  estimated  that  there  are  at  least 
20,000,000  stars  in  the  first  fourteen  magnitudes.     Those  stars 
which  are  visible  to  the  naked  eye  are  comprised  in  the  first  six 
magnitudes ;  and  stars  of  the  twentieth  magnitude  are  detected 
with  the  most  powerful  telescopes. 


CONSTELLATIONS.  215 

There  is,  however,  a  great  difference  in  the  brightness  of  stars 
which  belong  to  the  same  magnitude.  Sirius,  for  instance,  is 
fifteen  times  as  bright  as  some  other  stars  of  the  first  magnitude.* 

268.  Constellations. — In  order  to  facilitate  the  formation  of 
catalogues  of  the  stars,  they  are  separated  into  groups,  called 
constellations.     Ptolemy,  in  the  second  century,  enumerated  48 
constellations:  21  northern,  12  zodiacal,  and  15  southern.     The 
twelve  zodiacal  constellations  have  the  same  names  that  the  signs 
of  the  zodiac  bear,  which  are  given  in  Art.  91 ;  indeed,  the  signs 
really  took  their  names  from  the  constellations.     Owing,  how- 
ever, to  the  precession  of  the  equinoxes,  the  signs  and  the  con- 
stellations no  longer  coincide  (see  Art.  119),  the  constellation  of 
Aries  being  in  the  sign  of  Taurus,  &c. 

Since  the  time  of  Ptolemy,  about  sixty  other  constellations 
have  been  added  to  the  list.  Not  all  of  these,  however,  are  ac- 
cepted by  astronomers,  and  the  list  of  those  constellations  which 
are  generally  acknowledged  comprises  only  about  86  :  29  north- 
ern, 12  zodiacal,  and  45  southern. 

269.  Remarkable  Constellations. — The  most  remarkable  of  the 
northern  constellations  is  that  called  Ursa  Major,  or  the  Great 
Bear,  often  called  "  The  Dipper"  from  the  well-known  appearance 
presented  by  its  seven  conspicuous  stars.     The  two  of  these  seven 

*  Any  one  ?f  the  most  prominent  stars  may  be  identified  when  on  the 
meridian,  in  the  following  manner.  The  right  ascension  of  the  star,  given 
in  the  Ephemeris,  is,  by  Art.  9,  the  local  sidereal  time  of  the  star's  transit ; 
and  from  this  sidereal  time  the  local  mean  solar  time  can  be  obtained,  as 
proved  in  Art.  105,  by  subtracting  from  it  the  right  ascension  of  the  mean 
sun.  When  only  a  rough  estimate  is  desired,  the  quantity  given  on  page  1 I. 
of  each  month  in  the  Ephemeris,  in  the  last  column  on  the  right,  may  be 
taken  as  the  mean  sun's  right  ascension.  This  will  give  the  time  of  transit 
within  four  minutes.  The  more  rigorous  process  is  to  apply  to  the  quan- 
tity taken  from  page  II.  a  correction  taken  from  Table  III.  in  the  Appendix 
to  the  Ephemeris,  using  the  longitude  of  the  place  as  an  argument,  and 
adding  the  correction  if  the  longitude  is  west.  The  result  is  then  subtracted 
from  the  sidereal  time  as  above,  and  the  remainder  is  diminished  by  a 
correction  taken  from  Table  II.,  with  the  remainder  itself  as  an  argument. 
(See  example  in  the  latter  part  of  the  Ephemeris.) 

The  star's  meridian  altitude  is  found  by  the  method  given  in  the  note 
to  Art.  181. 


216  CONSTELLATIONS. 

stars  which  are  the  most  remote  from  the  handle  of  the  dipper 
are  called  the  pointers,  since  the  right  line  joining  them  will 
always,  when  prolonged,  pass  very  nearly  through  the  pole-star. 
This  constellation  contains  fifty-three  stars  of  the  first  five  magni- 
tudes, including  one  of  the  first  and  three  of  the  second. 

There  is  another  "Dipper,"  much  less  conspicuous,  consisting 
also  of  seven  stars,  the  pole-star  being  at  the  extremity  of  the 
handle.  These  stars  form  a  part  of  the  constellation  Ursa  Minor, 
which  contains,  in  all,  twenty-three  stars  of  the  first  five  magni- 
tudes. The  pole-star  itself  is  of  the  second  magnitude. 

The  constellation  Orion  is  a  magnificent  one,  and  was  fan- 
cifully supposed  by  the  ancients  to  bear  some  resemblance  to  a 
giant.  It  contains  two  stars  of  the  first  magnitude,  four  of  the 
second,  and  thirty-one  of  the  next  three  magnitudes.  The  three 
stars,  situated  nearly  in  a  straight  line,  which  form  the  giant's 
belt,  are  a  very  conspicuous  part  of  the  constellation.  This 
constellation,  in  the  northern  hemisphere,  bears  south  about  9  P.M. 
in  the  early  part  of  February ;  its  altitude  at  that  time  at  any 
place  being  nearly  equal  to  the  co-latitude  of  that  place.  Sirius, 
the  brightest  star  in  the  heavens,  is  also  seen  at  that  time  to  the 
left  of  this  constellation,  and  a  little  below  it. 

The  constellation  Pegasus  contains  forty-three  stars  of  the 
first  five  magnitudes.  Four  of  these  stars  are  of  the  second 
magnitude,  and  nearly  form  a  square.  This  square  bears  south 
about  9  P.M.  in  the  early  part  of  October,  with  an  altitude,  in 
the  northern  hemisphere,  about  15°  greater  than  the  co-latitude 
of  the  place  of  observation. 

The  constellation  Gemini,  or  the  Twins,  takes  its  name  from 
two  bright  stars,  nearly  of  the  first  magnitude,  called  Castor 
and  Pollux.  They  are  situated  near  each  other,  and  bear  south 
about  9  P.M.  in  the  latter  part  of  February,  in  the  northern 
hemisphere,  their  altitude  being  about  30°  greater  than  the  co- 
latitude  of  the  place  of  observation. 

270.  Stars  of  the  Same  Constellation. — Stars  of  the  same  con- 
stellation are  distinguished  from  each  other  by  the  letters  of  the 
Greek  or  the  Roman  alphabet,  or  by  numerals.  The  Greek 
letters  were  first  used  by  Bayer,  a  German  astronomer,  in  1604, 
who  called  the  brightest  star  in  a  constellation  a,  the  next 


NAMES    OF    STARS.  217 

brightest/3,  &c.  Thus  tne  poie-srar  bears  the  astronomical  name  of 
a  Ursx  Miiioris,  and  the  pointers  of  the  dipper  are  called  «  and 
p1  Ursse  Majoris.  Owing,  however,  either  to  carelessness  on  the 
part  of  Bayer  or  to  changes  in  the  brightness  of  some  of  the  stars, 
this  alphabetical  arrangement  does  not  in  all  cases  accurately 
represent  the  relative  brilliancy  of  the  stars  in  a  constellation. 

Tne  entire  number  of  stars  now  catalogued  amounts  to  several 
hundreds  of  thousands.  Three  catalogues  published  by  Arge- 
lander,  in  1859-62,  contain  over  320,000  stars  observed  at  Bonn. 
In  large  catalogues,  the  stars  are  usually  numbered  from  be- 
ginning to  end,  in  the  order  of  their  right  ascensions. 

271.  Stars  with  Special  Names. — Some  of  the  stars,  particularly 
the  more  conspicuous  ones,  have  special  names,  which  were  given 


*a  Eridani Achernar. 

a  Tauri Aldebaran. 

a  Aurigse Capella. 

(3  Orionis Rigel. 

*a  Argus Canopus. 

a  Canis  Majoris Sirius. 

a  Canis  Minoris Procyon. 

p  Geminorum Pollux. 

a  Leonis Regulus. 

a  Virginia Spica. 

a  Bootis Arcturus. 

a  Scorpii Antares. 

a  Lyrae Vega. 

a  Aquilae Altair. 

a  Piscis  Australis Fomalhaut. 


to  them  by  ancient  astronomers.  Instances  are  given  in  the 
accompanying  table,  all  the  stars  contained  in  it  being  commonly 
considered  to  be  of  the  first  magnitude.  About  90  stars  are 

thus  named,  though  most  of  these  names  are  no  longer  used. 
1!) 


218  CONSTITUTION    OF    STAHS. 

.All  of  these  stars,  excepting;  those  marked  with  an  asterisk, 
come  above  the  horizon  throughout  the  United  States.  The 
position  of  each,  in  right  ascension  and  declination,  can  be 
found  in  the  Ephemeris  for  any  day  in  the  year.  Besides  the 
stars  in  the  preceding  table,  there  are  three  others  of  the  first 
magnitude:  a  Crucis,  a  Centauri,  and  ft  Centauri.  They  are  all 
stars  of  large  southern  declination,  and  do  not  come  above  the 
horizon  in  any  part  of  the  United  States  excepting  the  southern 
parts  of  Texas  and  Florida. 

272.  Constitution  and  Diversity  of  Brightness. — The  spectra 
of  stars  are  found  to  contain  dark  lines,  similar  in  character  to 
those  by  which  we  have  seen  that  the  solar  spectrum  is  distin- 
guished (Art.  102).     These  systems  of  lines  differ  from  the  system 
of  lines  in  the  solar  spectrum,  and  they  are  also  different  in  dif- 
ferent stars.     This  difference,  however,  consists  in  the  absence 
of  certain  lines  seen  in  the  solar  spectrum,  and  not  in  the  pre- 
sence of  new  ones:    nearly   every    line    observed    having   its 
counterpart  in  the  solar  spectrum.     The  examination  of  these 
spectra,  and  the  comparison  of  the  dark  lines  which  they  con- 
tain with  the  bright  lines  found  in  the  spectra  of  terrestrial 
substances,  enable  us  to  establish   the  presence  of  certain  of 
these  substances  in  the  stars,  precisely  as  was  done  in  the  case 
of  the  sun.    In  Aldebaran,  for  instance,  the  presence  of  sodium, 
magnesium,  tellurium,  calcium,  antimony,  iron,  bismuth,  and 
mercury,  has  been  detected ;  in  Sirius,  of  sodium,  magnesium, 
iron,  and  hydrogen ;  in  a  Orionis,  of  magnesium,  sodium,  cal  - 
cium,  and  bismuth. 

The  diversity  of  brightness  in  the  stars  may  be  due  either  to 
a  difference  in  their  distances  from  us,  or  to  a  difference  in  their 
actual  magnitudes,  or  to  a  difference  in  the  intrinsic  splendor 
with  which  they  shine.  Probably  all  these  causes  exist ;  but  it 
is  fair  to  conclude  that,  as  a  general  rule,  the  brightest  stars  are 
the  nearest  to  us.  Observations  made  for  the  purpose  of  deter- 
mining the  distances  of  the  stars  go  to  justify  such  a  conclusion ; 
although  they  also  show  that  the  rule  is  not  an  absolute  one, 
since  some  of  the  fainter  stars  are  found  to  be  nearer  to  us  than 
some  of  the  brighter  ones. 

273.  Distance  of  the  Fixed  /Stars. — "N~o  perceptible  difference 


DISTANCE    OF    THE   STARS,  219 

is  detected  in  the  position  of  a  star  when  observed  at  places  of 
widely  different  latitudes.  The  conclusion  drawn  from  this 
fact  is,  that  the  stars  are  so  distant  that  lines  drawn  from  any 
two  points  on  the  earth's  surface  to  the  same  star  are  sensibly  pa- 
rallel :  in  other  words,  that  the  stars  have  no  geocentric  parallax. 
To  determine  the  distance  of  the  stars,  then,  we  must  have  re- 
course to  their  heliocentric  parallax.  In  Fig.  75  let  S  be  the 
sun,  AE'BE  the  orbit  of  the  earth,  s  the  position  of  a  fixed 


Fig.  75. 

star,  supposed  to  lie  in  the  plane  of  the  ecliptic,  and  NM  a  por- 
tion of  the  celestial  sphere.  From  s  draw  lines  sE,  sE',  tan- 
gent to  the  earth's  orbit,  and  also  prolong  them  beyond  s,  until 
they  meet  the  arc  of  the  celestial  sphere  NM.  Draw  the  radii 
vectores  SE'  and  SE  to  the  points  of  tangency.  If  we  suppose  the 
star  to  be  at  rest,  it  will  lie  in  the  direction  E's,  when  the  earth 
is  at  E',  and  in  the  direction  Es,  when  the  earth  is  at  E,  and 
the  motion  of  the  earth  about  the  sun  will  give  the  star  an  ap- 
parent oscillatory  movement  over  the  arc  ee'.  The  true  helio- 
centric direction  in  which  the  star  lies  is  Sa' ;  and  the  difference 
of  the  directions  in  which  the  star  lies  at  any  time  from  the  suri 
and  the  earth  is  its  heliocentric  parallax.  This  difference  of 
direction  is  evidently  at  its  maximum  when  the  earth  is  at  E'  or 
E;  and  this  maximum,  or  the  angle  SsE,  is  called  the  annual 
heliocentric  parallax,  or  simply  the  annual  parallax. 

Numerous  attempts  have  been  made  to  determine  the  annual 
parallax  of  the  stars,  by  comparing  observations  made  when  the 
earth  is  at  E  and  E'.  The  nicest  observation,  however,  has  failed 
to  detect  in  any  star  a  parallax  as  great  as  1";  and  in  only  12 
stars  has  the  slightest  appreciable  parallax  been  discovered. 


220  DISTANCE   OF   THE   STARS. 

274.  The  distance  of  the  fixed  stars,  then,  is  so  great  that  the 
radius  of  the  earth's  orbit,  92,400,000  miles,  does  not  subtend 
an  angle  of  even  I"  at  that  distance.     If,  in  the  triangle  SsE, 
•we  suppose  the  angle  SsE  to  be  equal  to  1",  we  shall  have, 

Sa  =  92,400,000  cosec  1": 

which  will  be  found  to  be  about  nineteen  trillions  of  miles.  This 
is  only  the  inferior  limit  of  the  distance  of  the  stars:  that  is  to 
say,  whatever  the  distance  may  be,  it  cannot  be  less  than  this ; 
but  how  much  greater  it  may  be,  particularly  in  the  case  of  those 
numerous  stars  in  which  no  movements  whatever  of  parallax 
can  be  detected,  it  is  impossible  to  calculate. 

275.  Immensity  of  this  Distance. — It  is  hardly  possible  to  obtain 
a  clear  conception  of  a  distance  of  nineteen  trillions  of  miles. 
Perhaps  the  nearest  approach  to  such  a  conception  is  made  by  con- 
sidering that  light,  moving  with  a  velocity  of  186,000  miles  a 
second,  and  passing  from  the  sun  to  the  earth  in  a  little  more  than 
eight  minutes,  would  consume  about  3^  years  in  accomplishing 
such  a  distance:  so  that  when  we  look  at  the  brightest  stars  in 
the  heavens,  we  see  them,  not  as  they  are  now,  but  as  they  were 
3£  years  ago;  and  if  any  one  of  them  were  to  be  destroyed  at 
any  instant,  we  should  continue  to  see  its  image  for  three  years 
and  more  after  that  time. 

Before  such  distances,  the  dimensions  of  the  solar  system 
shrink  to  the  insignificance  of  a  mere  point  in  space.  Neptune, 
the  most  distant  of  the  planets,  is  nearly  three  billions  of  miles 
from  the  sun ;  and  yet,  if  Neptune  and  the  sun  could  both  be 
seen  from  the  nearest  fixed  star,  the  angular  distance  between 
them  would  never  be  greater  than  about  30",  which  is  only  about 
6l0th  of  the  angle  which  the  sun  subtends  to  us. 

276.  Differential   Observations. — In    dealing  with  so  small  a 
quantity  as  the  annual  parallax  of  the  stars,  it  is  important  to 
avoid  all  circumstances  by  which  even  the  most  minute  errors 
may  be  entailed  upon  the  observations-.     The  apparent  position 
of  a  star  is  affected,  not  only  by  parallax,  but  by  precession, 
nutation,  aberration,  and   the  star's  own  proper  motion.     The 
laws  of  precession  and  nutation  enable  us  to  decide  what  amount 
of  the  apparent  motion  is  due  to  them;  and  the  effect  of  a  star's 
proper  motion  is  also  readily  separated  from  that  of  parallax, 


DISTANCE   OF   THE   STARS. 


221 


since  the  former  changes  the  position  of  the  star  from  year  to 
year,  while  the  latter  only  changes  its  position  during  the  year, 
causing  it  to  lie  now  on  one  side  and  now  on  the  other  of  its 
true  position,  but  giving  it  no  annual  progressive  motion.  But 
it  is  not  so  easy  to  separate  the  effects  of  aberration  and  paral- 
lax. Aberration,  as  we  have  already  seen  (Art.  125),  causes  a 
star  to  describe  a  circle,  an  ellipse,  or  an  arc,  about  its  true  po- 
sition as  a  centre,  according  to  its  situation  with  reference  to 
the  plane  of  the  ecliptic ;  and  it  is  easy  to  see  that  the  paral- 
lactic  movement  of  a  star  is  of  precisely  the  same  character. 
Thus  we  have  already  seen,  in  Fig.  75,  that  a  star  situated  in 
the  plane  of  the  ecliptic  will  oscillate  by  parallax  over  the  arc 
ee  ;  and  if  the  star  is  not  in  the  plane  of  the  ecliptic,  lines 
drawn  from  all  points  of  the  earth's  orbit  to  the  star,  and  thence 
prolonged  to  the  celestial  sphere,  will  evidently  meet  the  sphere 
in  a  circle  if  the  star  is  at  the  pole  of  the  ecliptic,  and  in  an 
ellipse  if  it  is  not  at  the  pole. 

In  order  to  separate  the  eifects  of  parallax  and  aberration, 
the  following  method  was  adopted  by  the  astronomer  Bessel. 
Instead  of  attempting  to  determine  by  direct  observation  the 
change  of  position  of  the  star  whose  parallax  was  sought,  he 
selected  another  star  of  much  less  magnitude,  and  therefore  sup- 
posed to  be  at  a  much  greater  distance,  which  lay  very  nearly  in 
the  same  direction  as  the  first  star,  and  observed  the  changes  in 
the  distance  between  these  two  stars  and  in  the  direction  of  the 
line  joining  them,  during  the  year.  Fig.  76  will  serve  to  ex- 
plain the  general  principle  of  this 
method.  Let  S  be  the  position  of  the 
star  whose  parallax  is  sought,  and  s 
the  position  of  the  smaller  star,  both 
being  projected  on  the  surface  of  the 
celestial  sphere.  By  the  motion  of  the 
earth  in  its  orbit  the  star  $  will  describe 
the  parallactic  ellipse  ADBC,  and  the 
star  s,  the  ellipse  adbc.  When  S  ap- 
pears to  be  at  A,  s  will  appear  to  be  at 
a;  when  S  is  at  D,  s  will  be  at  d,  &c. 
It  is  evident  that  Aa  and  Bb  will  lie 


222  MAGNITUDE   OF   THE    STARS. 

in  different  directions,  and  that  £d  will  be  greater  than  Cc:  and, 
therefore,  by  observing  the  different  directions  of  the  line  joining 
the  two  stars,  and  also  its  different  values,  during  the  year,  we 
may  obtain  the  difference  of  parallax  of  the  two  stars,  and,  ap- 
proximately, the  parallax  of  S. 

277.  Results. — This  method  was  applied  by  Bessel  to  the  star 
61  Cygni.     For  the  sake  of  greater  accuracy  he  made  use  of 
two  very  small  stars,  situated  very  near  to  that  star,  whose  ab- 
solute parallaxes  he  assumed  to  be  equal.    The  parallax  which  he 
obtained  for  this  star  was  0".35.     Other  observations  make  it 
0".56:  and  the  mean  of  these  two  values,  or  0".45,  corresponds 
to  a  distance  of  about  forty-two  trillions  of  miles,  a  distance 
which  light  would  require  seven  years  to  traverse.    The  parallaxes 
of  eleven  other  stars  have  been  obtained  with  more  or  less  accu- 
racy.   The  star  about  whose  parallax  there  is  the  least  doubt  is  a 
Centauri,  which  is  probably  the  nearest  to  us  of  all  the  stars.    Its 
parallax  is  0".92:  corresponding  to  a  distance  of  about  twenty- 
one  trillions  of  miles. ,  A  table  of  these  stars  is  given  at  the  end 
of  this  book. 

According  to  the  Russian  astronomer  Peters,  the  mean  paral- 
lax of  the  stars  of  the  first  magnitude  is  0".21,  corresponding  to 
a  distance  which  light  would  traverse  in  15s  years.  Another 
Kussian  astronomer,  Struve,  concludes  that  the  distance  of  the 
most  remote  stars  which  can  be  seen  in  Lord  Rosse's  great  tele- 
scope is  about  420  times  the  distance  of  the  stars  of  the  first 
magnitude:  from,  which  the  marvellous  inference  is  drawn  that 
the  distance  of  the  most  remote  telescopic  stars  from  the  earth  is 
only  traversed  by  light  in  6500  years. 

A  knowledge  of  the  distance  of  a  star,  and  of  its  proper  mo- 
tion, enables  us  to  estimate  the  amount  in  miles  of  its  transverse 
motion.  Sirius,  for  instance,  has  a  proper  motion  of  l".2o  a  year, 
and  its  parallax  is  0".23.  Hence  its  annual  transverse  motion  is 
equal  in  amount  to  the  radius  of  the  earth's  orbit  multiplied  by 
i^_5  .  which  is  a  motion  of  about  sixteen  miles  a  second.  This  is 
only  the  projection  upon  the  celestial  sphere  of  its  real  motion, 
which  may  be  much  greater. 

278.  Real  Magnitudes  of  the  Stars. — Hitherto,  when  we  have 
determined  the  distance  of  a  celestial  body,  we  have  been  able  to 


VARIABLE   STARS.  223 

compute  its  real  diameter  by  means  of  observations  made  upon 
its  angular  diameter.  But  this  method  fails  when  we  attempt  to 
make  use  of  it  in  obtaining  the  magnitude  of  the  stars,  since 
they  do  not  present  any  measurable  disc.  It  is  true  that  with 
the  better  class  of  telescopes  some  of  the  stars  appear  to  have  a 
sensible  disc;  but  this  disc  is  really  what  is  called  a  spurious 
one.  This  is  proved  by  the  fact  that  when  a  star  is  occulted  by 
the  moon,  the  size  and  the  shape  of  the  apparent  disc  remain  un- 
altered up  to  the  time  of  occultation,  and  its  disappearance  is  then 
instantaneous.  In  the  case  of  a  solar  eclipse,  however,  or  of  the 
occultation  of  a  planet,  the  disappearance  of  the  disc  is  gradual. 
We  can,  however,  obtain  some  idea  of  the  probable  magni- 
tude of  a  star  by  comparing  the  light  which  it  emits  with  that 
which  is  emitted  by  the  sun.  This  comparison  is  made  by 
means  of  the  light  of  the  moon.  The  ratio  of  the  light  of  the 
sun  to  that  of  the  full  moon,  and  the  ratio  of  the  light  of  the  full 
moon  to  that  of  some  of  the  stars,  have  been  obtained  by  appro- 
priate experiments;  and,  from  a  comparison  of  the  results  of 
these  experiments,  it  is  inferred  that  if  the  sun  were  removed  to 
a  distance  from  the  earth  equal  to  that  of  the  nearest  fixed  star, 
it  would  appear  only  as  a  star  of  the  second  magnitude.  The 
probability,  then,  is  that  unless  there  is  a  marked  difference  in 
the  intensity  of  the  light  which  these  different  bodies  emit,  the 
sun  is  not  so  large  as  most,  and  perhaps  all,  of  the  stars  of  the 
first  magnitude.  There  is,  however,  as  might  be  expected  from 
the  delicacy  of  the  observations,  some  discrepancy  in  the  results 
of  these  various  photometric  experiments :  the  light  of  Sirius,  for 
instance,  is  said  by  some  observers  to  be  one  hundred,  and  by 
others  to  be  four  hundred,  times  as  great  as  the  light  of  our  sun 
would  be,  were  it  removed  to  a  distance  from  us  equal  to  that 
of  Sirius. 

VARIABLE   AND   TEMPORARY   STARS. 

279.  Variable  Stars. — There  are  certain  stars  which  exhibit 
periodic  changes  in  their  brightness,  the  periods  being  in  some 
cases  only  a  few  days  in  length,  and  in  other  cases  embracing 
many  years.  The  star  o  Ceti,  called  also  Mir  a,  is  an  example 
of  this  class  of  stars.  AVhen  brightest,  it  is  a  star  of  the  second 


224  TEMPORARY   STARS. 

magnitude.  It  remains  in  this*  state  for  about  two  weeks,  and 
then  begins  to  diminish  in  brightness,  becoming  wholly  invisible 
to  the  naked  eye  in  about  three  months,  and  appearing  in  tele- 
scopes as  a  star  of  the  ninth  or  the  tenth  magnitude.  After  about 
five  months  it  again  appears,  and  in  three  months  again  reaches 
its  maximum  of  brightness.  The  period  in  which  these  changes 
occur  is  33H  days:  at  least,  that  is  its  mean  value:  its  extreme 
values  being  25  days  more  and  25  days  less  than  this.  It  is  also 
noticed  that  the  rate  of  its  increase  and  decrease  of  brightness  is 
not  always  the  same,  and  that  one  maximum  of  brightness  is 
not  always  equal  to  another. 

Algol)  or  ft  Persei,  is  another  remarkable  variable  star.  It 
remains  for  about  sixty-one  hours  as  a  star  of  the  second  magni- 
tude. At  the  end  of  that  time  it  begins  to  decrease  in  bright- 
ness, and  becomes  a  star  of  the  fourth  magnitude  in  less  than 
four  hours.  After  about  twenty  minutes  its  brightness  begins 
to  increase,  and  another  period  of  less  than  four  hours  brings  it 
up  again  to  a  star  of  the  second  magnitude. 

The  whole  number  of  variable  stars  is  between  150  and  200. 

280.  Several  theories  have  been  advanced  in  explanation  of 
this  periodicity  of  brightness  in  the  variable  stars.     One  theory 
is  that  the  surfaces  of  these  stars  are  not  uniformly  luminous,  and 
that  therefore,  in  rotating  upon  their  axes,  they  may  present  at 
one  time  the  lighter  portions  of  their  surfaces  to  the  earth,  and 
the  darker  portions  at  another.     Such  a  variation  of  luminosity 
might  be  caused  by  the  presence  of  spots  on  the  surfaces  of  the 
stars,  similar  to  the  spots  on  the  sun,  but  of  much  greater  extent. 
A  second  theory  is,  that  nebulous  bodies  may  revolve  as  satellites 
about  these  stars,  and  may  intercept  their  light  by  coming  in  be- 
tween them  and  the  earth.     The  irregularities  noticed  in  the  pe- 
riods of  these  stars,  however,  seem  to  constitute  an  objection  to  this 
second  theory,  while  they  may  be  allowed  by  the  first  theory,  since 
analogous  fluctuations  in  the  periods  and  the  magnitudes  of  the 
sun's  spots  have  been  observed.     Arago  suggests  that,  if  it  is 
true,  as  has  been  asserted  by  some  astronomers,  that  these  stars 
when  at  their  minimum  are  surrounded  by  a  kind  of  fog,  the 
diminution  of  light  may  be  due  to  the  interference  of  clouds. 

281.  Temporary  Stars. — There  are  stars  which  have  appeared 


DOUBLE   STARS.  225 

at  times  in  different  parts  of  the  heavens,  and  have  afterwards 
disappeared.  Such  stars  are  called  temporary  stars.  In  compar- 
ing recent  catalogues  of  stars  with  the  catalogues  of  ancient  as- 
tronomers, it  is  found  that  some  stars  which  were  formerly  visible 
are  no  longer  to  be  seen,  while  others  which  are  now  visible  to 
the  naked  eye  are  not  mentioned  in  the  ancient  catalogues. 
Some  of  these  cases  may  be  due  to  errors  of  observation,  but 
hardly  all  of  them.  Moreover,  similar  instances  have  occurred 
in  modern  times.  For  example,  it  is  recorded  by  the  Danish 
astronomer,  Tycho  Brahe,  that  in  November,  1572,  a  brilliant 
star  suddenly  blazed  forth  near  the  constellation  Cassiopeia,  and 
remained  in  sight  about  17  months.  When  at  its  brightest 
phase,  it  equalled  Venus  in  splendor,  and  was  visible  in  broad 
daylight.  It  disappeared  in  1574,  and  has  never  since  been 
seen.  Similar  temporary  stars  are  recorded  as  having  appeared 
in  or  near  the  same  place,  in  945  and  1264.  It  is  therefore 
possible  that  this  may  be  really  a  variable  star,  and  that  it  may 
reappear  in  the  latter  part  of  the  present  century. 

In  May,  1866,  a  star  of  the  ninth  magnitude,  in  the  constel- 
lation Corona  Borealis,  suddenly  increased  in  brightness,  and 
then  rapidly  decreased.  On  the  12th  of  the  month  it  was  of  the 
second  magnitude ;  on  the  14th,  of  the  third ;  and  the  rate  of 
decrease  in  its  brightness  was  for  some  time  about  half  a  mag- 
nitude a  day.  Lockyer  says  that  there  is  good  reason  to  believe 
that  this  sudden  increase  of  brilliancy  was  due  to  the  ignition 
of  hydrogen  in  the  star's  atmosphere. 

It  is  very  probable  that  these  temporary  stars  are  really  in 
no  respect  different  from  variable  stars,  except  in  the  length  of 
their  periods.  Sir  John  Herschel  says,  with  reference  to  these 
stars,  that  it  is  worthy  of  notice  that  all  of  them  which  are  on 
record  have  been  situated  in  or  near  the  borders  of  the  Milky 
Way. 

DOUBLE   AND    BINARY    STARS. 


12  Lyncis.         e  Lyrae. 

282.   Double  Stars. — Many  of  the  stars  which  appear  single 


226  .BINARY    STARS. 

to  the  naked  eye,  are  found,  "wjien  examined  in  telescopes,  to 
consist  of  two  stars,  apparently  very  near  to  each  other.  These 
are  called  double  stars.  Only  four  were  known  to  exist  until  near 
the  close  of  the  last  century,  when  Sir  William  Herschel  dis- 
covered about  500.  The  whole  number  of  double  stars  now 
known  exceeds  9000.  In  some  cases  the  two  stars  are  nearly 
of  the  same  magnitude,  but  more  frequently  one  of  them  is  a 
large  star,  and  the  other  a  small  one.  Castor  is  an  instance  of 
the  former  class,  and  Sirius,  Vega,  and  the  pole-star  are  instances 
of  the  latter  class.  Some  stars  are  found  to  consist  of  three,  four, 
five,  or  more  stars,  and  are  called  triple,  quadruple,  &c.  stars. 
The  star  e  Lyrse,  for  instance,  appears  in  ordinary  telescopes  to 
consist  of  two  stars,  but  with  telescopes  of  greater  power  each 
of  these  stars  is  resolved  into  two  others. 

283.  Binary  Stars. — The  question  arises  with  reference  to  these 
combinations :  are  the  stars  which  compose  them  really  con- 
nected, as  are  the  sun  and  the  planets ;  or  is  their  appearance 
merely  an  optical  illusion,  arising  from  the  fact  that"  the  stars 
in  any  one  combination  happen  to  lie  in  the  same  direction 
from  the  earth,  although  they  may  at  the  same  time  be  at  an 
immense  distance  from  each  other?  The  chances,  at  all  events, 
are  very  much  against  the  latter  supposition.  The  astronomer 
Struve  has  calculated  that  there  is  only  about  one  chance  in  9570 
that,  if  the  stars  of  the  first  seven  magnitudes  were  scattered  at 
random  in  the  heavens,  any  two  of  them  would  fall  within  4" 
of  each  other;  and  yet  more  than  100  such  cases  have  been 
observed.  He  has  further  calculated  that  there  is  only  about  one 
chance  in  200,000  that  three  stars  would  accidentally  fall  within 
30"  of  each  other,  so  as  to  form  a  triple  star;  and  at  least  four 
such  cases  are  to  be  found. 

The  chances,  then,  are  that  the  stars  in  these  various  combi- 
nations are  physically  connected.  But  more  than  this :  it  was 
announced  by  Sir  William  Herschel  in  1803,  after  twenty-five 
years  of  observation  upon  many  of  the  double  stars,  that  in 
each  double  star  which  he  had  examined,  the  two  stars  of  which 
it  was  composed  revolved  about  each  other  in  regular  orbits, 
and  in  fact  constituted  a  sidereal  system.  Subsequent  observa- 
tions bj  other  astronomer?  have  fully  verified  this  conclusion, 


BINARY    STARS.  227 

anrl  about  600  double  stars  have  been  found  to  consist  of  stars 
revolving  about  each  other,  or  rather  about  their  common  centre 
of  gravity,  according  to  the  Newtonian  law  of  gravitation. 
Such  double  stars  are  called  binary  stars,  to  distinguish  them 
from  other  double  stars,  the  components  of  which  have  not  as 
yet  been  found  to  be  physically  connected.  There  are  also 
other  double  stars,  the  components  of  which,  while  as  yet  they 
do  not  seem  to  revolve  about  each  other,  have  constantly  the 
same  proper  motion  :  thus-  showing  that  they  are  in  all  probability 
moving  as  one  system  through  space.  Triple,  quadruple,  &c. 
stars,  whose  constituents  are  found  to  be  physically  connected, 
are  called  ternary,  quaternary,  &c.  stars. 

The  orbits  of  the  binary  stars  are  found  to  be  ellipses  of  con- 
siderable eccentricity.  The  periods  of  their  revolutions  have 
also  been  approximately  determined,  and  extend  over  a  very 
wide  range.  There  are  only  about  twelve  stars  whose  periods 
are  less  than  100  years,  and  only  about  150  whose  periods  are 
less  than  1000  years. 

284.  Alpha  Centauri. — The  star  a  Centauri,  a  star  of  the  first 
magnitude  in  the  southern  hemisphere,  is  found  to  consist  of 
two  components,  one  of  the  first  magnitude  and  the  other  of  the 
second.  The  relative  positions  of  these  two  components  have 
been  carefully  noted  durirg  the  last  40  yeacs. 
In  Fig  77,  A  represents  one  of  those  compo- 
nents, and  B  B  B"  the  apparent  path  of  the 
other  about  it.  The  major  axis  of  the  orbit 
is  about  40'',  and  the  period  75  years.  Its 
eccentricity  is  0.63. 

Since  the  distance  of  a  CeiJtauri  from  the 
earth  is  approximately  known,  we  can  ob- 
tain some  idea  of  the  dimensions  of  the 
orbit.  If  E  in  the  figure  represents  the  earth, 
we  shall  have,  in  the  right-angled  triangle 
CO  E,  the  angle  E  equal  to  20",  and  the  side  ? 

EO  equal  to  21  trillions  of  miles.     Hence,  Fig.  77. 

CO  =  OEx  tan  20"  =  2,000,000,000  miles  : 
which  is  about  equal  to  the  distance  of  Uranus  from  the  sun,  or 
to  22  times  the  radius  of  the  earth's  orbit. 


228 


COLORED    STATES. 


Again,  knowing  the  radius  o/  the  orbit  and  the  period,  we 
can  obtain  an  approximate  value  of  the  mass  of  a  Centauri 
from  the  formula  in  Art.  214.  It  will  be  found  to  be  about 
twice  the  mass  of  the  sun. 

COLORED    STARS. 

285.  Many  of  the  stars,  both  isolated  and  double,  shine  with 
a  colored  light.  The  isolated  colored  stars  are  usually  red  or 
orange :  blue  or  green  stars  being  very  uncommon.  The  num- 
ber of  red  stars  now  recognized  is  at  least  300,  about  40  of 
which  are  visible  to  the  naked  eye.  Among  the  most  con- 
spicious  of  the  red  stars  are  Aldebaran  and  Antares ;  and  it  is 
worthy  of  notice  that  nearly  all  the  variable  stars  are  of  this 
color.  According  to  Mr.  Ennis's  observations,  Capella,  Kigel, 
Procyon,  and  Spica  are  blue;  Sirius,  Vega,  and  Altair  are  green; 
and  Arcturus  is  yellow. 

The  components  of  the  double  stars  are  often  of  different  co- 
lors; blue  and  yellow,  or  green  and  yellow;  and,  less  frequently, 
white  and  purple,  or  white  and  red.  The  following  table  con- 
tains a  few  of  the  many  instances  of  such  stars  which  might  be 
given : — 


Name. 

Magnitude 
of 
Components. 

Color  of  the  Larger. 

Color  of  the  Smaller. 

7}  Cassiopeiae  
a  Piscium  

4  7 
5  6 

Yellow. 
Pale  Green. 

Purple. 
Blue. 

i  Cancri  
e  Bootis  
C  Coronae  
pCygni  

5  8 
3  7 
5  6 
3  7 

Orange. 
Pale  Orange. 
White. 
Yellow. 

Blue. 
Sea  Green. 
Purple. 
Blue. 

When  the  colors  of  the  components  are  complementary,  and  the 
components  are  of  very  unequal  size,  it  is  possible  that  only  one 
of  the  colors  may  really  exist;  the  other  being,  according  to  a 
law  of  Optics,  merely  the  result  of  contrast.  Such  an  opinion  is 
held  by  some  astronomers ;  but  the  objection  is  raised  to  it  by 
others  that,  if  one  of  these  colors  is  only  accidental,  it  ought  to 
disappear  when  the  eye  is  shielded  from  the  light  of  the  star 


PLATE  IV. 


NEBULJE. 

1.  ANNULAR  NEBULA,  57  M  LYR/€. 

2.  PLANETARY  NEBULA,  3614  H  VIRGINIS. 

3.  NEBULOUS  STAR,  i  ORIONIS. 

4.  SPIRAL  NEBULA,  99  M  VIRGINIS. 

5.  CRAB  NEBULA  IN  TAURUS. 


NEBULAE.  229 

which  has  the  other  color:  which,  however,  is  very  far  from 
Jteing  the  case.  Another  objection  is  that  a  similar  phenomenon 
ought  to  be  seen  in  all  colored  stars  whose  components  are  of 
miaqual  sizes:  whereas  many  double  stars  are  found  in  which 
both  components  have  the  same  color,  sometimes  red  and  some- 
times blue.  In  one  of  Struve's  catalogues,  out  of  596  double 
stars,  there  were: — 

295  pairs,  both  white: 
118  pairs,  both  yellowish  or  reddish: 
63  pairs,  both  bluish : 
120  pairs  of  totally  different  colors. 

In  a  few  instances  stars  have  changed  their  color.  Sirius, 
for  instance,  is  now  green;  but  both  Ptolemy  and  Seneca  ex- 
pressly state  that  in  their  day  it  had  a  reddish  hue.  Capella, 
which  is  now  blue,  was  formerly  red.  Some  observers  state  that 
seventeen  stars  of  the  first  magnitude  are  colored,  and  that  seven 
of  these  have  changed  their  color.  This  change  of  color  is  par- 
ticularly .noticeable  in  the  variable  stars.  In  that  of  1572  (Art. 
281),  the  color  changed  from  white  to  yellow,  and  then  to  a  de- 
cided red;  and  it  is  observed  generally  in  the  variable  stars  that 
the  redness  of  the  light  increases  as  its  intensity  diminishes. 


CLUSTERS   AND    NEBULAE. 

286.  If  we  examine  the  heavens  on  any  clear  night  when  the 
moon  is  below  the  horizon,  we  shall  find  here  and  there  groups 
of  stars,  which  present  a  hazy,  cloud-like  appearance.  These 
groups  are  classified  into  clusters  and  nebulce,  although  it  is  diffi- 
cult to  establish  any  precise  distinction  between  the  two  classes. 
When  the  different  members  of  a  group/  or  at  all  events  some 
of  them,  can  be  separated  from  each  other  with  the  naked  eye,  the 
group  is  called  a  cluster.  A  nebula,  on  the  other  hand,  is  either 
wholly  invisible  to  the  naked  eye,  or,  if  seen  at  all,  presents,  as 
the  name  implies,  only  an  ill-defined,  cloudy  appearance.  The 
number  of  nebulae  which  have  been  discovered  is  over  5000. 
Some  of  these  nebulae  are  resolved,  with  the  aid  of  the  telescope, 
into  separate  stars:  while  others,  even  when  examined  in  the 
most  powerful  telescopes,  preserve  their  cloudy  appearance,  and 


230 

give  no  trace  of  stars.  The  former  are  called  resolvable  nebuja;, 
the  latter  irresolvable  nebulae. 

As  every  increase  of  telescopic  power  has  rendered  resolvable 
some  of  the  nebulse  previously  classified  as  irresolvable,  the  com- 
mon opinion  of  astronomers,  until  within  the  last  few  years,  was 
that  there  was  no  distinction  between  these  nebulse  more  funda- 
mental than  that  of  distance,  magnitude,  or  intensity  of  light. 
Recently,  however,  the  light  of  some  of  these  nebulas  has  been 
subjected  to  spectroscopic  analysis,  and  the  results  seem  to  show 
the  existence  of  decidedly  different  classes  of  nebulse.  The  re- 
searches of  Mr.  Huggins  show  that  the  light  of  some  of  these 
nebulse  gives  spectra,  which  are  distinguished,  not  by  dark  lines, 
but  by  bright  ones :  which  shows,  as  we  have  already  noticed 
(Art.  102),  that  the  light  does  not  come  from  solid  bodies  in  a 
state  of  ignition,  but  from  incandescent  vapor  or  gas.  Some  at 
least,  then,  of  the  nebulse  are  to  be  considered  as  bodies  of  a 
gaseous  nature ;  and  there  are  indications  that  the  gases  of  which 
they  are  composed  are  hydrogen  and  nitrogen.  The  Great  Ne- 
bula in  Orion  is  an  instance  of  this  class  of  nebulse. 

Th(>  nebulse  are  very  unequally  distributed  over  the  heavens. 
They  congregate  especially  in  a  zone  crossing  the  Milky  Way  at 
right  angles,  and  they  are  especially  abundant  in  the  constella- 
tions Leo,  Ursa  Major,  and  Virgo. 

287.  Examples  of  Clusters. — The  most  noticeable  cluster  is  the 
well-known  group  called  the  Pleiades.  This  group  contains  six 
or  seven  stars  which  are  visible  to  the  naked  eye,  the  brightest 
star  being  of  the  third  magnitude :  and  glimpses  of  many  more 
can  be  obtained  by  examining  the  group  with  the  eye  turned 
sideways.  With  the  aid  of  a  telescope  between  one  hundred  and 
two  hundred  stars  can  be  detected. 

The  Hyades  are  a  group  in  Taurus,  near  the  star  Aldebaran,  in 
which  from  five  to  seven  stars  can  be  counted.  This  group  was 
supposed  by  the  ancients  to  have  some  influence  upon  the  rain. 

Prsesepe,  or  the  Bee-hive,  in  Cancer,  is  visible  to  the  naked  eye 
as  a  luminous  spot.  With  a  telescope  of  moderate  power,  more 
than  forty  conspicuous  stars  are  seen  within  a  space  about  30'  square, 
together  with  many  smaller  ones.  Before  the  invention  of  the  te- 
lescope, this  group  was  probably  one  of  the  few  recognized  nebulse. 


NEBULAE.  231 

288.  Different  Forms  of  Nebulce. — The  groups  which  usually 
go  by  the  name  of  nebulae  present,  as  might  be  expected,  almost 
every  variety  of  form ;  and  most  of  these  forms  are  too  irregular 
to  admit  of  any  classification  or  description.    Such  is  not  the  case, 
however,  with  all  of  them :  and  some  of  the  different  varieties  of 
shape  are  exemplified  in  the  following  list: — 

1.  Annular  Nebulae; 

2.  Elliptic  Nebulas; 

3.  Spiral  Nebulise; 

4.  Planetary  Nebulas; 

5.  Nebulous  stars. 

There  are  also  individual  nebulae  whose  names  indicate  the 
general  appearance  which  they  present :  such  are  the  "  Horse-shoe 
Nebula,"  the  "Crab  Nebula,"  the  "Dumb-bell  Nebula,"  &c. 

289.  Annular  Nebulae. — Of  annular    or  ring-shaped  nebulae, 
the  heavens  present  four  examples.    The  most  remarkable  one  is 
situated  in  the  northern  constellation  Lyra.     Sir  John  Herschel 
says  of  it:   "It  is  small,  and   particularly  well-defined,  so   as 
to  have  more  the  appearance  of  a  flat,  oval,  solid  ring  than  of*  a 
nebula.     The  axes  of  the  ellipse  are  to  each  other  in  the  pro- 
portion of  about  four  to  five,  and  the  opening  occupies  rather 
more  than  half  the  diameter.     The  central  vacuity  is  filled  iu 
with  faint  nebulas,  like  a  gauze  stretched  over  a  hoop.     The 
powerful  telescopes  of  Lord  Rosse  resolve  this  object  into  ex- 
cessively minute  stars,  and  show  filaments  of  stars  adhering  to 
its  edges." 

290.  Elliptic  Nebulce. — There  are  several  instances  of  elliptic 
or  oval  nebulas,  of  various  degrees  of  eccentricity.     The  very 
conspicuous  nebula  called  "  the  Great  Nebula  in  Andromeda"   is 
an  example  of  this  class.     This  object  is  distinctly  visible  to  the 
naked  eye,  and  is  sometimes  mistaken  for  a  comet.    When  viewed 
with  a  telescope  of  moderate  power,  it  has  an  elongated  oval 
form ;  but  in  the  largest  telescopes  its  aspect  is  very  different 
from  this.     Professor  G.  P.  Bond  traced  it  through  a  length  of 
4°,  and  a  breadth  of  2^°,  and  also  discovered  in  it  two  curious 
black  streaks,  lying  nearly  parallel  to  the  major  axis  of  the 
oval.     He  also  succeeded  in  detecting  in  it  evidence  of  a  stellar 
constitution. 


232  NEBULAE. 

Some  elliptic  nebulae  are  remarkable  for  having  double  stars 
at  or  near  each  of  their  foci. 

291.  Spiral  Nebulas. — Observations  with  the  Earl  of  Rosse's 
telescope   have   led  to  the  discovery  of  nebulae  which  consist 
of  spiral  bands  proceeding  from  a  common  nucleus,  and  some- 
times from  two  nuclei.     The  most   remarkable  of  these  spiral 
nebulae  is  situated  near  the  extremity  of  the  tail  of  the  Great 
Bear.     It  consists  of  nebulous  coils  diverging  from  two  luminous 
centres,  about  5'  apart,  and  gives  evidence  of  stellar  composition. 

292.  Planetary  Nebulce. — Planetary  nebulae  received  their  name 
from  Sir  William  Herschel,  on  account  of  the  resemblance  in  form 
which  they  bear  to  the  larger  planets  of  our  system.     The  out- 
lines of  some  of  them  are  well  defined,  while  those  of  others  are 
indistinct;  and  some  of  them  shine  with  a  blue  light.     About 
twenty-five  have  been  discovered,  most  of  them  being  situated 
in  the  southern  hemisphere.     One  of  these  nebulae  is  situated 
near  ft  Ursse  Majoris,  and  has  a  diameter  of  2'  40".     It  was  dis- 
covered by  Me*ehain,  in  1781,  and  is  described  as  "a  very  singular 
object,  circular  and  uniform,  which  after  a  long  inspection  looks 
like  a  condensed  mass  of  attenuated  light."     Perforations  and  a 
spiral  tendency  have  been  detected  in  it  by  the  Earl  of  Rosse. 

These  planetary  nebulae  can  hardly  be  globular  clusters  of 
stars,  as  they  would  in  that  case  be  brighter  in  the  middle  than 
at  the  borders,  instead  of  being,  as  they  are,  uniformly  bright 
throughout.  Some  astronomers  suppose  that  they  are  assem- 
blages of  stars,  arranged  either  in  cylindrical  beds,  or  in  the 
form  of  hollow  spherical  shells. 

293.  Nebulous  Stars. — Nebulous  stars  are  stars  which  are  sur- 
rounded by  a  faint  nebulous  envelope,  usually  circular  in  form, 
and  sometimes  several  minutes  in  diameter.     In  some  cases  this 
envelope  terminates  in  a  distinct  outline,  while  in  others  it  gra- 
dually fades  away  into  darkness.     According  to  Hind,  nebulous 
stars  "have  nothing  in  their  appearance  to  distinguish  them  from 
others  entirely  destitute  of  such  appendages ;  nor  does  the  nebu- 
lous matter  in  which  they  are  situated  offer  the  slightest  indica- 
tions of  resolvability  into   stars,  with  any  telescopes   hitherto 
constructed." 

294.  Double  Nebulce. — M.  D'Arrest,  of  Copenhagen,  mentione 


VARIATION   OF  BRIGHTNESS.  233 

fifty  double  nebulae  whose  components  are  not  more  than  5'  apart, 
and  estimates  that  there  may  be  as  many  as  200  such  double 
nebulse.  The  two  components  are  generally  of  a  globular  form. 
It  is  possible  that  these  components  may  in  some  cases  be 
physically  connected.  In  one  instance  there  seem  to  have  been 
changes  in  the  distance  and  the  relative  position  of  the  two  mem- 
bers during  the  last  eighty  years,  which  indicate  the  possibility 
of  a  revolution  of  one  of  the  members  about  the  other. 

295.  Magellanic  Clouds. — These  two  nebulse,  called  also  nube- 
cula  major  and  nubecula  minor,  are  situated  near  the  southern 
pole  of  the  heavens.     They  are  visible  to  the  naked  eye  at  night, 
when  the  moon  is  not  shining,  and  have  an  oval  shape.     One 
of  them  covers  an  area  of  forty-two  square  degrees,  the  other  an 
area  of  ten  square  degrees.     Sir  John  Herschel  found  them  to 
consist    of    swarms   of    stars,    clusters,   and   nebulse   of    every 
description. 

296.  Variation  of  Brightness  in  the  Nebulce. — In  the  case  of  a 
few  of  the  nebulse  a  change  of  brightness  has  been  discovered, 
which  is  to  some  extent  analogous  to  what  has  already  been 
noticed  in  connection  with  the  variable  and  the  temporary  stars. 
In  1852,  Mr.  Hind  discovered  a  small  nebula  in  the  constella- 
tion Taurus.     This  nebula  was  observed  from  that  time  until 
the  year  1856;  but  in  1861  it  had  entirely  disappeared.     In 
1862  it  was  seen  in  the  telescope  at  Pulkowa,  and  for  a  short 
time  appeared  to  be  increasing  in  brightness;  but  towards  the 
end  of  1863,  Mr.  Hind,  upon  searching  for  it  with  the  telescope 
with  which  it  was  originally  discovered,  was  unable  to  find  any 
trace  of  it.     It  is  a  curious  coincidence  that  a  star  situated  very 
near  to  this  nebula  has  changed,  in  the  same  interval  of  time, 
from  the  tenth  magnitude  to  the  twelfth. 

There  are  four  or  five  instances  on  record  of  similar  changes 
in  the  brightness  of  nebulse.  The  nebula  known  as  80  of 
Messier's  Catalogue,  is  said  to  have  changed,  in  the  months 
of  May  and  June,  1860,  from  a  nebula  to  a  well-defined  star 
of  the  seventh  magnitude,  and  then  to  have  resumed  its  ori- 
ginal appearance.  The  change  was,  however,  so  rapid  and  so 
decided,  that  some  astronomers  are  inclined  to  ascribe  it  to  the 
existence  of  a  variable  star  between  the  nebula  and  the  earth, 


234  MILKY   WAY. 

rather  than  to  a  variation  in  tflie  nebula  itself.  The  Great  Ne- 
bula in  the  constellation  Argo,  when  observed  by  Sir  John  Her- 
schel in  1838,  contained  in  its  densest  part  the  star  y,  which  was 
then  of  the  first  magnitude.  Observations  made  in  1863,  how- 
ever, showed  that  this  star  had  become  entirely  free  from  its 
nebulous  envelope,  and  had  also  been  reduced  to  a  star  of  the 
sixth  magnitude.  The  outline  of  certain  parts  of  the  nebula 
was  also  found  to  be  different  from  what  it  had  been  represented 
to  be  by  Herschel.  More  recent  observations  show  still  further 
changes  in  this  nebula. 

297.  The  Galaxy,  or  Milky  Way.— The  Milky  Way,  that  well- 
known  luminous  band  which  stretches  through  the  heavens,  may 
be  considered  to  be  a  continuous  resolvable  nebula.     If  we  follow 
the  line  of  its  greatest  brightness,  and  neglect  occasional  devia- 
tions, we  find  its  course  to  be  nearly  that  of  a  great  circle  of  the 
heavens,  inclined  to  the  equinoctial  at  an  angle  of  about  63°. 
At  one  point  a  kind  of  branch  is  sent  off,  which  unites  again 
with  the  main  stream  at  a  distance  of  about  150°  from  the  point 
of  separation.     The  angular  breadth  of  the  belt  varies  from  6° 
to  20°,  the  average  breadth  being  about  10°.     When  examined 
with  the  telescope,  this  band  is  found  to  be  made  up  of  thousands 
and  millions  of  telescopic  stars,  grouped  together  with  every 
degree  of  irregularity.     Of  the  20,000,000  stars  of  the  first  four- 
teen magnitudes,  about  nine-tenths  are  in  or  near  the  Milky  Way. 
At  some  points  the  stars  are  so  close  to  each  other  that  they 
form  one  bright  mass  of  light;  while  at  other  points  there  are 
dark  spaces  containing  scarcely  a  star  which  is  visible  to  the 
naked  eye.     A  very  noticeable  break  of  this  kind  occurs  in  that 
part  of  the  Milky  Way  which  lies  nearest  to  the  south  pole. 
This  dark  space  is  *about  8°  in  length  and  5°  in  breadth.     It 
contains  only  one  star  visible  to  the  naked  eye,  and  that  is  a  very 
small  one.     Early  southern  navigators  gave  to  this  vacancy  the 
name  of  the  Coal  /Sack. 

298.  Number  of  the  Stars. — Scarcely  more  can  be  said  of  the 
probable  number  of  the  stars,  than   that  it  is  as  inconceivably 
great  as  are  the  distances  by  which  they  are  separated  from  the 
earth  and  from  each  other.     Sir  William  Herschel  states  that 
on  one  occasion,  while  observing  the  stars  in  the  Milky  Way,  he 


MOTION   OF   THE   SOLAR   SYSTEM.  235 

estimated  that  116,000  stars  passed  through  the  field  of  his  tele- 
scope in  fifteen  minutes;  and  that,  on  another  occasion,  nearly 
260,000  stars  passed  in  forty-one  minutes.  It  may  assist  us  in 
correctly  appreciating  the  magnitude  of  these  numbers  to  re- 
member that  the  number  of  stars  visible  to  the  naked  eye  in 
the  whole  heavens  is  less  than  7000  (Art.  266).  Results  still 
more  astonishing  follow  from  an  examination  of  the  nebulae. 
Take,  for  instance,  the  "  Great  Nebula  in  Andromeda,"  which, 
by  observations  made  with  the  telescope  at  Cambridge,  Mas- 
sachusetts, within  the  last  few  years,  has  been  found  to  give 
evidence  of  stellar  composition.  According  to  Professor  G.  P. 
Bond,  its  length  is  4°,  and  its  breadth  2J°.  Now,  if  a  telescope 
just  suffices  to  resolve  a  nebula  into  separate  stars,  it  is  safe  to 
assume  that  the  angular  distance  between  any  two  contiguous 
stars  in  it  cannot  be  greater  than  1".  If,  then,  we  multiply  to- 
gether the  number  of  seconds  in  the  length  of  this  nebula,  and 
the  number  of  seconds  in  its  breadth,  we  obtain  over  129,000,000 
for  an  approximate  estimate  of  the  number  of  stars  in  this  one 
nebula. 

MOTION   OF    THE   SOLAR   SYSTEM    IN   SPACE. 

299.  We  have  now  accustomed  ourselves  to  consider  the  earth 
to  be  a  planet,  rotating  upon  a  fixed  axis  once  in  every  sidereal 
day,  and  revolving  about  the  sun  once  in  every  sidereal  year;  but 
we  have  up  to  this  point  considered  the  sun  and  the  solar  sys- 
tem to  be  at  rest  in  space.  There  are,  however,  observations 
which  tend  to  show  that  such  is  not,  in  truth,  the  case,  but 
that  the  whole  solar  system  is  in  rapid  motion  through  space. 
Analogy  certainly  supports  such  a  conclusion.  "VVe  have  already 
seen  that  the  solar  system,  immense  as  it  may  seem  to  be  in 
itself,  must  be  considered  as  scarcely  more  than  a  point  in  com- 
parison with  the  entire  celestial  system  (Art.  275);  and  we 
have  also  seen  reasons  for  concluding  (Art.  278)  that  there  are 
many  bodies  among  the  stars  which  are  very  much  larger 
than  the  sun.  There  is,  then,  no  reason  for  supposing  that  the 
solar  system  is  the  centre  of  the  celestial  system;  and  there  is 
nothing  violent  in  the  conclusion  that,  just  as  Jupiter  revolves 
about  the  sun,  and  carries  its  satellites  with  it,  so  the  sun  may 


236 


MOTION   OF   THE   SOLAR   SYSTEM. 


revolve  about  some  other  body  or'point,  and  carry  its  system  of 
planets  and  satellites  with  it. 

300.  The  Apparent  Motiom  of  the  Stars. — The  proper  motions 
of  the  stars,  already  mentioned,  may  be  explained  in  three  dis- 
tinct ways.  First,  they  may  be  what  they  seem  to  be,  real 
motions  of  the  stars,  performed  in  orbits  of  immense  radii;  or, 
secondly,  they  may  be  only  apparent  motions,  caused  by  a  change 
of  the  position  of  the  solar  system  in  space;  or,  thirdly,  they 
may  be  the  results  of  both  these  causes  existing  together.  The 
last  of  these  theories  was  advanced  by  Sir  William  Herschel, 
in  1783 ;  and  although  doubts  were  afterwards  expressed  as  to 
its  truth,  later  observations  have  tended  to  support  it,  and  it  is 
now  generally  accepted  by  astronomers.  If  we  suppose  for  a 
moment  that  the  solar  system  is  approaching  any  given  point 
in  the  heavens,  the  pole-star,  for  instance,  the  result  will  be  that 
the  stars  which  lie  about  that  point  will  appear  to  separate 
slowly  from  it,  while  the  stars  which  lie  about  the  diametrically 


opposite  point  will  appear  to  close  up  around  that  point ;  and  in- 
deed all  the  stars  will  apparently  move  in  an  opposite  direction 
to  that  in  which  we  suppose  the  solar  system  to  be  moving,  those 


MOTION    OF    THE   SOLAR   SYSTEM.  237 

stars  having  the  greatest  motion  which  lie  in  a  direction  at  right 
angles  to  the  direction  of  the  motion  of  the  solar  system.  In 
Fig.  78,  let  E  be  the  position  in  space  of  the  earth  at  any  time, 
and  let  P,A,B,D,  and  G  be  stars  supposed  to  be  situated  at  equal 
distances  from  the  earth.  Let  the  earth,  by  the  motion  of  the 
whole  solar  system  in  space,  be  carried  to  some  point  E',  in  the 
direction  of  the  star  P.  It  is  evident  that  the  angular  distance 
of  the  star  A  from  P,  when  the  earth  is  at  E',  or  the  angle  PE'A, 
is  greater  than  the  angular  distance  between  A  and  P  when  the 
earth  is  at  E,  or  the  angle  PEA.  In  other  words,  while  the 
earth  has  moved  from  E  to  E',  the  star  A  has  apparently  receded 
from  P.  So  also  the  star  D  has  apparently  approached  the  star 
C.  The  stars  B  and  G  have  botli  receded  from  P,  the  retro- 
gradation  of  G,  or  the  angle  EGE',  being  greater  than  the 
retrogradation  of  B,  or  the  angle  EBE'. 

301.  Direction  and  Amount  of  this  Motion. — The  elements, 
then,  of  the  motion  of  the  solar  system  through  space  are  deter- 
mined from  what  are  called  the  proper  motions  of  the  stars, 
llecent  observations  and  calculations  have  not  only  confirmed 
Herschel's  theory  that  such  a  motion  really  exists,  but  have  also 
very  nearly  confirmed  his  conclusion  as  to  the  point  towards  which 
the  motion  is  directed.  Different  astronomers  give  different 
positions  to  this  point;  but  they  all  agree  in  the  general  conclu- 
sion, that  the  solar  system  is  at  present  moving  in  the  direction 
of  a  point  in  the  constellation  Hercules,  situated  in  about  17£ 
hours  of  right  ascension,  and  35°  of  north  declination. 

The  calculations  of  M.  Otto  Struve  lead  him  to  the  conclu- 
sion that  for  a  star  situated  at  right  angles  to  the  direction  of 
the  motion  of  the  solar  system  and  at  the  mean  distance  of  the 
stars  of  the  first  magnitude,  the  annual  angular  displacement 
of  the  star  due  to  that  motion  is  0."34 :  that  is  to  say,  the  dis- 
tance through  which  the  system  moves  in  one  year  subtends  at 
the  star  an  angle  of  that  amount.  Now,  the  mean  parallax  of 
the  stars  of  the  first  magnitude,  or  the  angle  subtended  at  the 
mean  distance  of  those  stars  by  the  radius  of  the  earth's  orbit,  is 
estimated  tobeO."21  (Art.  277);  hence  the  annual  motion  of  the 
solar  system  is  the  radius  of  the  earth's  orbit  multiplied  by  Jf, 
or  about  150,000,000  miles:  a  little  less  than  five  miles  a  second. 


238  ORBIT   OF   THE   SOLAR   SYSTEM. 

This  is  the  estimate  commonly  accepted  by  astronomers.  Mr. 
Airy,  however,  deduces  a  motion  of  about  twenty -seven  miles  a 
second. 

302.  The  Orbit  of  the  Solar  System. — Although  the  motion  of 
the  solar  system  through  space  appears  to  be  rectilinear,  it  does 
not  follow  that  such  is  actually  the  case:  since  it  may  move  in 
an  elliptical  orbit,  with  a  radius  vector  so  immense  that  years 
and  even  centuries  of  observation  will  be  needed  to  show  any 
sensible  change  of  direction.     The  probability  is  that  the  sun, 
with  its  planets  and  their  satellites,  revolves  about  the  common 
centre  of  gravity  of  the  group  of  stars  of  which  it  is  a  member; 
and  that  this  centre  of  gravity  is  situated  in  or  near  the  plane 
of  the  Milky  Way.     If  we  suppose  the  orbit  in  which  our 
system  is  moving  to  be  an  ellipse  with  a  small  eccentricity,  then 
the  centre  of  this  ellipse  will  lie  in  a  direction  which  makes  an 
angle  of  about  90°  with  the  direction  in  which  the  system  is  now 
moving.     Miidler  concluded  that  Alcyone,  the  brightest  star  in 
the  Pleiades,  was  the  central  sun  of  the  celestial  sphere,  while 
Argelander,  believing  that  this  central  point  must  lie  in  the 
principal  plane  of  the  Milky  Way,  places  it  in  the  constellation 
Perseus.     It  may  be  noticed,  in  connection  with  this  subject,  that 
Sir  William  Herschel  was  inclined  to  believe  that  some  of  the 
more  conspicuous  of  the  stars,  such  as  Sirius,  Arcturus,  Capella, 
Vega,  and  others,  were  in  a  great  degree  out  of  the  reach  of 
the  attractive  power  of  the  other  stars,  and  were  probably,  like 
the  sun,  centres  of  systems. 

303.  Years,  and,  in  all  probability,  ages  of  observation  will 
be  needed  to  determine  the  elements  of  this  most  stupendous 
orbit.     Madler's  speculations  have  led  him  to  the  conclusion 
that  the  period  of  this  revolution  is  29,000,000  years :  a  period 
in  comparison  with  which  the  years  through  which  astronomical 
observations  have  extended  are  utterly  insignificant.     The  stu- 
dent who  is  curious  to  know  more  of  this  subject  will  find  the 
details  fully  set  forth  in  Grant's  History  of  Physical  Astronomy. 
Grant  himself  says,  in  reference  to  the  subject: — "It  is  manifest 
that  al\  such  speculations  are  far  in  advance  of  practical  astro- 
nomy, and  therefore  they  must  be  regarded  as  premature,  how- 
ever probable  the  supposition  on  which  they  are  based,  or  how- 


REAL    MOTIONS   OF   THE   STARS.  239 

ever  skilfully  they  maybe  connected  with  the  actual  observation 
of  astronomers." 

DETERMINATION    OF   THE   REAL   MOTIONS   OF   THE   STARS. 

304.  We  have  already  seen  (Art.  265),  that  in  order  to  deter- 
mine the  real  motion  of  a  star  in  space,  we  must  be  able  to 
determine,  not  only  its  transverse  motion,  which  is  indicated  by 
a  change  in  its  apparent  position  upon  the  celestial  sphere,  but 
also  its  motion  directly  to  or  directly  from  the  earth.     Now,  a 
star  situated  at  the  nearest  distance  of  the   fixed   stars,  and 
moving  towards  the  earth  with  a  velocity  equal  to  that  of  the 
earth  in  its  orbit  (18  miles  a  second),  would  diminish  its  dis- 
tance from  us  by  only  about  -3^th  in  a  thousand  years.     The 
detection  of  any  such  motion  by  a  change  in  the  star's  apparent 
brightness  is,  therefore,  utterly  out  of  the  question.     The  spectro- 
scope, however,  gives  us  quite  another  means  of  detecting  such 
a  motion. 

305.  Analogy  Between  Light  and  Sound. — A  clear  conception 
of  the  principle  upon  which  this  method  rests  may  be  more 
readily  obtained  if  we  first  notice  the  analogy  between  the  con- 
duction of  light  and  that  of  sound.     Sound  is  the  result  of  a 
series  of  waves  or  pulses  in  the  air,  produced  by  the  vibrations 
of  a  sonorous  body ;  light  is  the  result  of  a  similar  series  of 
pulses  or  waves  in  the  luminiferous  ether,  caused  by  the  vibra- 
tions of  the  particles  of  a  luminous  body.     The  more  rapidly 
a  sonorous  body  vibrates,  the  greater  will  be  the  number  of 
pulses  or  waves  which  it  communicates  to  the  air  in  the  unit  of 
time,  and,  consequently,  the  higher  will  be  the  pitch  of  the  tone 
produced.     In   the  case  of  a  luminous  body,  the  greater  the 
number  of  waves  in  the  luminiferous  ether  which  the  vibrations 
of  any  particle  cause  in  the  unit  of  time,  the  greater  will  be  the 
refrangibility  of  the  ray  produced ;  in  the  solar  spectrum,  for 
instance,  the  violet  rays  are  the  most  refrangible  of  all  the  rays 
which  we  can  see,  and  the  number  of  waves  in  the  unit  of  time 
necessary  to  produce  them  is  greater  than  the  number  necessary 
to  produce  rays  of  any  other  color.     The  refrangibility  of  a  ray 
is  therefore  analogous  to  the  pitch  of  a  tone. 

306.  Change,  of  Tone  or  Pefrnngibilihj. — It  is  proved  by  ex- 


240  REAL   MOTIONS   OF   THE   STARS. 

periment  that  if  the  distance  between  a  sonorous  body,  producing 
a  tone  of  constant  pilch,  and  the  hearer,  is  diminished  by  the 
motion  of  either,  with  a  velocity  that  has  an  appreciable  ratio  to 
that  of  sound,  the  pitch  of  the  tone  will  appear  to  be  heightened  ; 
and  that  if,  on  the  contrary,  the  distance  between  the  two  is  in- 
creased with  sufficient  rapidity,  the  pitch  of  the  tone  will  appear 
to  be  lowered.  So,  too,  in  the  case  of  light:  if  the  luminous 
body  and  the  observer  approach  each  other,  the  refrangibility 
of  the  rays  of  light  will  be  increased ;  and  if  they  separate,  it 
will  be  diminished.  If,  then,  we  can  detect  any  change  of  re- 
frangibility in  the  light  of  a  heavenly  body,  we  may  conclude 
that  the  distance  of  that  body  from  the  earth  is  changing. 

307.  Change  of  Refrangibility  Detected. — Father  Secchi,  in  a 
communication  to  the  Comptes  Rendus  in  the  early  part  of  1868, 
after  stating  the  principles  of  this  method,  announced  that  he 
had  subjected  the  light  of  Sirius  and  of  other  prominent  stars  to 
an  examination  with  the  spectroscope,  but  had  detected  no  evi- 
dence of  motion.  Since  then,  Mr.  Huggins,  an  English  observer, 
who  has  devoted  himself  especially  to  spectroscopic  investiga- 
tions, has  succeeded  in  detecting  such  a  motion  in  Sirius.  There 
is  a  certain  dark  line  F  in  the  blue  of  the  solar  spectrum,  which 
corresponds  to  a  bright  line  in  the  spectrum  of  hydrogen  ;  and 
this  same  line  also  appears  in  the  spectrum  of  Sirius.  Now, 
if  Sirius  has  no  motion  either  towards  or  from  the  earth,  the 
line  F  in  its  spectrum  will  coincide  in  position  with  the  corres- 
ponding line  in  the  spectrum  of  hydrogen,  when  the  two  spectra 
are  compared  by  means  of  the  spectroscope:  while  if,  on  the 
contrary,  Sirius  has  such  a  motion,  its  whole  spectrum,  lines 
and  all,  will  be  shifted  bodily,  and  the  line  F  will  no  longer 
coincide  with  the  corresponding  line  in  the  spectrum  of  hydrogen. 

Mr.  Huggins,  using  a  spectroscope  of  large  dispersive  power, 
and  carefully  comparing  the  spectrum  of  Sirius  with  that  of 
hydrogen,  finds  that  the  line  F  in  the  spectrum  of  Sirius  is 
displaced,  by  about  ^th  of  an  inch.  This  displacement  is 
towards  the  red  end  of  the  spectrum,  and  indicates  that  the 
refrangibility  of  the  light  of  Sirius  is  diminished :  since  the 
red  rays  are  the  least  refrangible  of  all  the  colored  rays  of  the 
spectrum.  Sirius  is  therefore  receding  from  the  earth. 


REAL    MOTIONS    OF    THE   STARS.  241 

308.  Amount  of  the  Real  Motion  of  Sirius. — The  amount  of 
recession  corresponding  to  a  displacement  of  this  extent,  when 
observed  in  a  spectroscope  whose  power  is  equal  to  that  of  the 
one  used  by  Mr.  Huggins,  is  computed  to  be  about  41  j  miles 
a    second.     But   it   happens   that   when   the    observation    was 
made,  the  earth  was  so  situated  in  its  orbit  that  it  was  receding 
from  Sirius,  by  its  revolution  in  its  orbit,  at  the  rate  of  about 
12  miles  a  second.     The  motion  of  the  solar  system  in  space, 
computed   to  be  five  miles  a  second  (Art.  301),  must  also   be 
taken   into  consideration ;    and  the  point  towards   which   this 
motion  is  at  present  directed  is  almost  exactly  opposite  to  the 
position  of  Sirius  on  the  celestial  sphere.     The  earth  was  there- 
lore  moving  away  from  Sirius  at  the  rate  of  about  17  miles  a 
second.     If,  then,  we  diminish  the  whole  amount  of  the  increase 
of  distance  between  Sirius  and  the  earth  in  one  second,  by  the 
amount  of  the  motion  of  the  earth  through  space  in  one  second, 
or  17  miles,  we  find  that  Sirius  is  moving  through  space,  away 
from  the  earth,  at  the  rate  of  about  24A  miles  a  second. 

By  combining  this  motion  with  the  transverse  motion  of 
Sirius,  we  can  obtain  an  approximate  value  of  its  real  motion. 
In  Art.  277,  the  transverse  motion  of  Sirius  was  computed  to 
be  about  16  miles  a  second.  The  resultant  of  these  two  motions 
is  a  motion  of  about  29  miles  a  second :  or,  900,000,000  miles 
a  year. 

Mr.  Huggins  further  concludes  that  five  stars  of  the  Dipper 
are  receding  from  us,  and  that  the  bright  star  Arcturus  is  ap- 
proaching us. 

309.  The  numerical  results  of  the  preceding  article  may  not 
be  beyond  criticism  ;  but  the  grand  fact  remains,  that  in  all 
probability  the  motions  of  these  distant  bodies,  which  have  so 
long  seemed  to  be  wrapped  in  hopeless  mystery,  are  soon  to 
come  within  the  reach  of  our  observation.     The  knowledge  of 
these  motions  will  be  a  powerful  auxiliary  in  the  determination 
of  the  law  which  undoubtedly  binds  all  the  heavenly  bodies 
together  in  one  great  system  ;    and  it  is  not  presumptuous  to 
expect  that  at  some  future  day — no  one  can  say  how  distant  or 
how  near — this  law  will  be  revealed  to  us. 

21 


242  APPENDIX. 


APPENDIX. 


MATHEMATICAL  DEFINITIONS  AND   FORMULA. 

PLANE   TRIGONOMETRY. 

1.  The  complement  of  an  angle  or  arc  is  the  remainder  ob- 
tained by  subtracting  the  angle  or  arc  from  90°. 

2.  The  supplement  of  an  angle  or  arc  is  the  remainder  ob- 
tained by  subtracting  the  angle  or  arc  from  180°. 

3.  The  reciprocal  of  a  quantity  is  the  quotient  arising  from 

dividing  1  by  that  quantity :  thus  the  reciprocal  of  a  is  — • 

4.  In  the  series  of  right  triangles  ABC,  AB'C',  AB"C",  &c., 

Fig.  79,  having  a  common  angle  A,  we  have  by 

Geometry, 

BC  __  B'C'  _  B"C" 
AB  ~  AB'  ~  =  AB"' 
BC  __  B'C'  B"C" 
AC~  AC'  r~  AC"' 

41L—4K  _  AB" 

~AC~  AC'  ~~  AC*' 

The  ratios  of  the  sides  to  each  other  are  there- 
Fig.  79.  £ore  ^ne  same  jn  an  right  triangles  having  the 
same  acute  angle :  so  that,  if  these  ratios  are  known  in  any  one 
of  these  triangles,  they  will  be  known  in  all  of  them.  These 
ratios,  being  thus  dependent  only  on  the  value  of  the  angle, 
without  any  regard  to  the  absolute  lengths  of  the  sides,  have 
received  special  names,  as  follows  : 

The  sine  of  the  angle  is  the  quotient  of  the  opposite  side 

BC 
divided  by  the  hypothenuse.     Thus,  sin  A  =  -T-TV 

The  tangent  of  the  angle  is  the  quotient  of  the  opposite  side 

BC 

divided  by  the  adjacent  side.     Thus,  tan  A  = 

-A  \j 


TRIGONOMETRY.  243 

The  secant  of  the  angle  is  the  quotient  of  the  hypothenuse 
divided  by  the  adjacent  side.    Thus,  sec  A  = 


5.  The  cosine,  cotangent,  and  cosecant  of  the  angle  are  respec- 
tively the  sine,  tangent,  and  secant  of  the  complement  of  the 
angle.  Now,  in  Fig.  79,  the  angle  ABC  is  evidently  the  com- 
plement of  the  angle  BA  C.  Hence  we  have, 

.    p       AC 
cos  A  =  sin  B  =  -T-=  ; 


AC 
BC 
AB 


cot  A  =  tan  B  =  j^^, 

cosec  A  —  sec  B  = 

6.  Since  the  reciprocal  of  y-  is  — ,  we  see,  from  the  preceding 

o        a 

definitions,  that  the  sine  and  the  cosecant  of  the  same  angle,  the 
tangent  and  the  cotangent,  the  cosine  and  the  secant,  are  re- 
ciprocals. 

7.  If  two  parts  of  a  plane  right  triangle  in  addition  to  the  right 
angle  are  given,  one  of  them  being  a  side,  the  triangle  can  be 
solved:  that  is  to  say,  the  values  of  the  remaining  parts  can  be 
obtained.     This  solution  is  effected  by  means  of  the  definitions 
above  given. 

8.  When  an  angle  is  very  small,  its  sine  and  its  tangent  are 
both  very  nearly  equal  to  the  arc  which  subtends  the  angle  in 
the  circle  whose  radius  is  unity.     Hence,  to  find  the  sine  or  the 
tangent  of  a  very  small  angle  approximately,  we  have,  if  a;  is  a 
small  angle  expressed  in  seconds, 

sin  x  =  tan  x  =  x  sin  1". 
If  x  is  expressed  in  minutes, 

sin  x  =  tan  x  =  x  sin  1'. 

9.  If  x  and  y  are  any  two  small  angles,  we  have  from  the  pre- 
ceding article, 

sin  x x  sin  1"    _x  . 

sin  y       y  sin  V       y  ' 

that  is,  the  sines  (or  tangents)  of  very  small  angles  are  propor- 
tional to  the  angles  themselves. 

10.  Cos  x  =  1  —  2  sm2  J  x. 


244 


APPENDIX. 


11.  The  sides  of  a  plane  triangle  are  proportional  to  the  sinea 
of  their  opposite  angles. 

12.  In  order  to  solve  a  plane  oblique  triangle,  three  parts  must 
be  given,  and  one  of  them  must  be  a  side. 

SPHERICAL   TRIGONOMETRY. 

13.  A  spherical  triangle  is  a  triangle  on  the  surface  of  a  sphere, 
formed  by  the  arcs  of  three  great  circles  of  the  sphere. 

14.  In  a  spherical  right  triangle,  the  sine  of  either  oblique 
angle  is  equal  to  the  quotient  of  the  sine  of  the  opposite  side, 
divided  by  the  sine  of  the  hypothenuse.     Thus,  in  the  triangle 

B       ABC,  right-angled  at  C,  Fig.  80,  we  have, 

sin  EC 
sin  AB ' 

15.  The  cosine  of  either  oblique  angle  is 
equal  to  the  quotient  of  the   tangent  of  the 
adjacent  side,  divided  by  the  tangent  of  the 
Thus, 

tan  A  C 


sin  A  =. 


Fig.  80. 


hypothenuse. 


cos  A  = 


tan  AB 


16.  The  tangent  of  either  oblique  angle  is  equal  to  the  quo- 
tient of  the  tangent  of  the  opposite  side,  divided  by  the  sine  of 
the  adjacent  side.     Thus, 

tan  BC 

tan  A  —  -. T-^  • 

sin  AC 

17.  A  spherical  right  triangle  can  be  solved  when  any  two 
parts  in  addition  to  the  right  angle  are  given.     The  solution  is 
effected  by  means  of  the  formula}  given  in  the  preceding  articles. 

ANALYTIC   GEOMETRY. 

18.  The  circle,  the  ellipse,  the  hyperbola,  and  the  parabola  are 

often  called  conic  sections;  since  each 
curve  may  be  formed  by  intersecting 
a  right  cone  by  a  plane. 

19.  An  ellipse  is  a  plane  curve,  in 
which  the  sum  of  the  distances  of  each 
point  from  two  fixed  points  is  equal  to 
a  given  line.  Thus,  in  Fig.  81,  the 


ANALYTIC   GEOMETRY. 


245 


sum  of  the  distances  of  the  point  M  from  the  two  fixed  points  F 
and  F'  is  always  constant,  wherever  on  the  curve  ACBD  the 
point  M  may  be  situated. 

The  two  fixed  points  are  called  the  foci  of  the  ellipse,  and  the 
middle  point  of  the  line  which  joins  them  is  called  the  centre. 

A  line  drawn  from  either  focus  to  any  point  of  the  curve  is 
called  a  radius  vector. 

A  line  drawn  through  the  centre,  and  terminating  at  each  ex- 
tremity in  the  curve,  is  called  a  diameter.  That  diameter  which 
passes  through  the  foci  is  called  the  transverse  or  major  axis: 
and  that  diameter  drawn  perpendicular  to  the  transverse  axis  is 
called  the  conjugate  or  minor  axis.  Thus,  AB  is  the  transverse 
axis,  and  CD  the  conjugate  axis.  The  transverse  axis  is  equal 
to  the  constant  sum  of  the  distances  FM  and  F'M. 

The  eccentricity  of  the  ellipse  is  the  quotient  of  the  distance 
from  the  centre  to  either  focus,  divided  by  half  the  major  axis. 

OF 

Thus,  -y=  is  the  eccentricity. 

20.  The  solid  generated  by  the  revolution  of  an  ellipse  about 
its  major  axis  is  called  a  prolate  spheroid:  that  generated  by  its 
revolution  about  its  minor  axis  is  called  an  oblate  spheroid. 

The  expression  for  the  volume  of  an  oblate  spheroid  is  ^xa^b : 
in  which  a  and  b  denote  the  semi-major  and  the  semi-minor  axis 
of  the  generating  ellipse,  and  x  the  ratio  of  the  circumference 
of  a  circle  to  its  diameter,  or  3.1416. 

The  compression,  or  oblateness,  of  an  oblate  spheroid  is  the 
ratio  of  the  difference  between  the  major  and  the  minor  axis  of 
the  generating  ellipse  to  the  major  axis. 

21.  The  hyperbola  is  a  plane  curve,  in  which  the  difference  of 
the  distances  of  each  point 

from  two  fixed  points  is 
equal  to  a  given  line. 
Thus,  in  Fig.  82,  the  dif- 
ference of  the  distances  of 
any  point  M  of  the  curve 
from  the  two  fixed  points 
F  and  F'  is  equal  to  a 
given  line. 


246 


APPENDIX. 


The  two  fixed  points  are  called  the  foci,  and  the  middle  point 
of  the  line  joining  them  is  called  the  centre. 
22.  The  parabola  is  a  plane  curve,  every 
point  of  which   is  equally  distant   from  a 
fixed  point,  and  from  a  right  line  given  in 
the  plane.     Thus,  in  Fig.  83,  in  which  AB 
is  the  given  line,  and  F  the  given  point,  the 
distances  FM  and    GM  are  equal  to  each 
other,  for  any  point  M  of  the  curve. 
Fig.  ss.  The  fixed  point  is  called  the  focus. 


MECHANICS. 

23.  The  resultant  of  two  or  more  forces  is  a  force  which  singly 
produce  the  same  mechanical  effect  as  the  forces  themselves 

jointly. 

The  original  forces  are  called  components.  In  all  statical  in- 
vestigations the  components  may  be  replaced  by  their  resultant, 
and  vice  versa. 

24.  If  two  forces  be  represented  in  magnitude  and  direction 
by  the  two  adjacent  sides  of  a  parallelogram,  the  diagonal  will 
represent  their  resultant  in  magnitude  and  direction.     Thus,  in 

Fig.  84,  if  a  force  act  at  A 
in  the  direction  AD',  and  a 
second  force  act  at  A  in  the 
direction  AB',  these  two  forces 
being  represented  in  magni- 
tude by  the  lengths  of  the 
lines  AD  and  AB  respectively, 
the  resultant  of  these  two 
forces  will  be  a  force  in  the 
direction  A  Cf,  and  of  a  magnitude  represented  by  the  length 
of  the  line  AC.  The  parallelogram  is  called  the  parallelogram 
of  forces. 

25.  Conversely:    any  given  force  may  be  resolved  into  two 
component  forces,  acting  in  given  directions.     Thus,  in  Fig.  84, 
let  A  C  be  the  given  force,  and  AB'  and  AD'  the  given  directions. 
From  C  draw    CB  parallel  to  AD' }  and   CD  parallel  to  AB'. 


247 

AB  and  AD  will  be  the  two  components  acting  in  the  given 
directions. 

26.  The  force  which  must  continually  urge  a  material  point 
towards  the  centre  of  a  circle,  in  order  that  it  may  describe  the 
circumference  with  a  uniform  velocity,  is  equal  to  the  square  of 
the  linear  velocity  divided  by  the  radius.  This  proposition  is 
expressed  in  the  following  formula  ; 


v  being  the  space  passed  over  in  the  unit  of  time,  r  the  radius  of 
the  circle,  and  /  the  magnitude  of  the  force. 

27.  The  force  which  constantly  urges  a  body  towards   the 
centre  of  its  circular  path  is  called  a  centripetal  force.     The 
tendency  which  the  body  has  to  recede  from  the  centre,  in  con- 
sequence of  its  inertia,  or  the  resistance  which  it  offers  to  a  de- 
flection from  a  rectilinear  path,  the  resistance  being  estimated 
in  the  direction  of  the  radius,  is  called  a  centrifugal  force.     These 
two  forces  are  in  equilibrium  as  long  as  the  body  moves  in  the 
same  circular  path,  and  the  same  expression  serves  for  the  mea- 
sure of  each  force. 

28.  Let  t  be  the  periodic  time,  or  the  time  of  one  revolution. 
We  shall  evidently  have, 

vt  =  2-r; 


Substituting  this  value  of  v2  in  the  formula  in  Art.  26  above,  we 
shall  have, 


KIRKWOOD'S  LAW. 

In  1848,  Daniel  Kirkwood,  of  Pennsylvania,  announced  the 
discovery  of  an  analogy  in  the  distances,  masses,  periods  of  ro- 
tation, and  periods  of  revolution  of  the  principal  planets.  This 
analogy  is  known  under  the  name  of  Kirkwood' s  Law.  The 


248  APPENDIX. 

original  statement  of  it  will  be  found  in  the  American  Journal 
of  Science,  Vol.  ix.,  Second  Series,  and  is  as  follows: 

"  Let  P  be  the  point  of  equal  attraction  between  any  planet 
and  the  one  next  interior,  the  two  being  in  conjunction:  P'  that 
between  the  same  and  the  one  next  exterior. 

"Let  also  D  =  the  sum  of  the  distances  of  the  points  P,  P' 
from  the  orbit  of  the  planet,  which  I  shall  call  the  diameter 
of  the  sphere  of  the  planet's  attraction  : 

"  jy  =  the  diameter  of  any  other  planet's  sphere  of  attraction 
found  in  like  manner: 

"n  =  the  number  of  sidereal  rotations  performed  by  the 
former  during  one  revolution  around  the  sun: 

"  nf  =  the  number  performed  by  the  latter  ;  then  it  will  be 
found  that 

n2  :  n/f  =  D5  :  D'3  ;  or  n  =±  ri     —, 


That  is,  the  square  of  the  number  of  rotations  made  by  a  planet 
during  one  revolution  round  the  sun,  is  proportional  to  the  cube 

n 
of  the  diameter  of  its  sphere  of  attraction:  or  j^  is   a  constant 

quantity  for  all  the  planets  of  the  solar  system." 

This  analogy,  when  first  announced,  created  much  interest 
among  scientific  men,  many  of  whom  considered  that,  if  its  truth 
were  established,  it  would  be  a  powerful  argument  in  favor  of  the 
nebular  hypothesis.  There  is  so  much  uncertainty  attending  the 
determination  of  the  masses  and  the  periods  of  rotation  of  many 
of  the  planets,  that  the  statement  of  this  analogy  has  been  ex- 
cluded from  the  main  text  of  this  book. 


ASTRONOMICAL  CHRONOLOGY. 

The  science  of  Astronomy  seems  to  have  been  cultivated  in 
very  early  ages  by  almost  all  the  Eastern  nations,  particularly 
by  the  Egyptians,  the  Persians,  the  Indians,  and  the  Chinese. 
Records  of  observations  made  by  the  Chaldseans  as  far  back  as 
2234  B.C.  are  said  to  have  been  found  in  Babylon.  The  study 
of  Astronomy  was  continued  by  the  Chaldseans,  and  in  later  times 


ASTRONOMICAL  CHRONOLOGY.  249 

by  the  Greeks  and  the  Romans,  until  about  200  A.D.  After  that 
time  it  was  neglected  for  about  six  centuries,  and  was  then  re- 
sumed by  the  Eastern  Saracens  after  Bagdad  was  built.  It  was 
brought  into  Europe  in  the  thirteenth  century  by  the  Moors  of 
Barbary  and  Spain.  A  full  account  of  the  rise  and  the  progress 
of  the  study  of  Astronomy,  is  given  in  Vinee's  Astronomy, 
Vol.  IT. 

The  instrument  principally  used  by  the  ancient  astronomers 
seems  to  have  been  a  sort  of  quadrant,  mounted  in  a  vertical 
position.  Ptolemy  describes  one  which  he  used  in  determining 
the  obliquity  of  the  ecliptic.  The  Arabian  astronomers  had  one 
with  a  radius  of  21  f  feet,  and  a  sextant  with  a  radius  of  57 f  feet. 

The  following  dates,  with  scarcely  any  change  or  addition, 
have  been  taken  from  George  F.  Chambers's  Descriptive  Astro- 
nomy (Oxford,  1867),  in  which  many  other  interesting  astrono- 
mical dates,  here  omitted,  may  be  found^  [See  Note,  page  254.] 

B.C. 

720.  Occurrence  of  a  lunar  eclipse,  recorded  by  Ptolemy.  It 
seems  to  have  been  total  at  Babylon,  March  19,  9ih.  P.M. 

640-550.  Thales,  of  Miletus,  one  of  the  seven  wise  men  of  Greece. 
He  declared  that  the  earth  was  round,  calculated  solar 
eclipses,  discovered  the  solstices  and  the  equinoxes,  and  re- 
commended the  division  of  the  year  into  365  days. 

585.  Occurrence  of  a  solar  eclipse,  said  to  have  been  predicted 
by  Thales. 

580-497.  Pythagoras,  of  Crotona,  founder  of  the  Italian  Sect. 
He  taught  that  the  planets  moved  about  the  sun  in  elliptic 
orbits,  and  that  the  earth  was  round;  and  is  said  to  have 
suspected  the  earth's  motion. 

547.  Anaximander  died.  He  asserted  that  the  earth  moved,  and 
that  the  moon  shone  by  light  reflected  from  the  sun.  He 
introduced  the  sun-dial  into  Greece. 

500.  Parmenides,  of  Elis,  taught  the  sphericity  of  the  earth. 

490.  Alcmreon,  of  Crotona,  asserted  that  the  planets  moved  from 
west  to  east. 

450.  Diogenes,  of  Apollonia,  stated  that  the  inclination  of  the 
earth's  orbit  caused  the  change  of  seasons.  Anaxagoras  is 
said  to  have  explained  eclipses  correctly. 


250  APPENDIX. 

B.C. 

432.  Melon  introduced  the  luni-solar  period  of  19  years.  He 
observed  a  solstice  at  Athens  in  424. 

384-322.  Aristotle  wrote  on  many  physical  subjects,  including 
Astronomy. 

370.  Eudoxus  introduced  into  Greece  the  year  of  3651  days. 

330.  Callippus  introduced  the  luni-solar  period  of  76  years. 
Pytheas  measured  the  latitude  of  Marseilles,  and  showed 
the  connection  between  the  moon  and  the  tides.  He  also 
showed  the  connection  of  the  inequality  of  the  days  and 
nights  with  the  differences  of  climate. 

320-300.  Autolychus,  author  of  the  earliest  works  on  Astronomy 
extant  in  Greek. 

306.  First  sun-dial  erected  in  Rome. 

287-212.  Archimedes  observed  solstices,  and  attempted  to  mea- 
sure the  sun's  diameter.  Aristarchus  declared  that  the 
earth  revolved  about  the  sun,  and  rotated  on  its  axis. 

276-196.  Eratosthenes  measured  the  obliquity  of  the  ecliptic, 
and  found  it  to  be  20  2  degrees.  He  also  measured  a  degree 
of  the  meridian  with  considerable  exactness. 

190-120.  Hipparchus,  called  the  Newton  of  Greece.  He  dis- 
covered precession :  used  right  ascensions  and  declinations, 
and  afterwards  latitudes  and  longitudes :  calculated  eclipses : 
discovered  parallax :  determined  the  mean  motions  of  the  sun 
and  the  moon;  and  formed  the  first  regular  catalogue  of  the 
stars. 

50.  Julius  Caesar,  with  the  Egyptian  astronomer  Sosigenes,  under- 
took to  reform  the  calendar. 

A.D. 

100-170.  Ptolemy,  of  Alexandria,  author  of  the  Ptolemaic  Sys- 
tem of  the  Universe,  in  which  the  earth  is  the  centre.  He 
wrote  descriptions  of  the  heavens  and  the  Milky  Way, 
and  formed  a  catalogue  giving  the  positions  of  1022  fixed 
stars.  He  appears  to  have  been  the  first  to  notice  the  re- 
fraction of  the  atmosphere. 

642.  The  School  of  Astronomy,  at  Alexandria,  founded  ten 
centuries  previously  by  Ptolemy  the  Second,  was  destroyed 
by  the  Saracens  under  Omar. 

762.  Rise  of  Astronomy  among  the  Eastern  Saracens. 


ASTRONOMICAL    CHRONOLOGY.  251 

A.n. 
880.  Albatani,  the  most  distinguished  astronomer  between  Hip- 

parchus  and  Tycho  Brahe.  He  corrected  the  values  of  pre- 
cession and  the  obliquity  of  the  ecliptic,  formed  a  catalogue 
of  the  stars,  and  first  used  sines,  chords,  &c. 

1000.  Abul-Wefa  first  used  secants,  tangents,  and  cotangents. 

1080.  The  use  of  the  cosine  introduced  by  Geber,  and  also  some 
improvements  in  Spherical  Trigonometry. 

1252.  Alphonso  X.,  King  of  Castile,  under  whose  direction  cer- 
tain celebrated  astronomical  tables,  called  the  Alphomine 
Tables,  were  compiled. 

1484.  Waltherus  used  a  clock  with  toothed  wheels.  (The  earliest 
complete  clock  of  which  there  is  any  certain  record  was 
made  by  a  Saracen  in  the  thirteenth  century.) 

1543.  Publication  of  Copernicus's  De  Revolutionibus  Orbium  Ce- 
lestium,  setting  forth  the  Copernican  System  of  the  Universe. 

1581.  Galileo  determined  the  isochronism  of  the  pendulum. 

1582.  Tycho  Brahe  commenced  astronomical  observations  near 
Copenhagen. 

1594.  Gerard  Mercator,  author  of  Mercator's  Projection.     (The 

date  is  doubtful,  and  may  have  been  as  early  as  1556.) 
1576.  Fabricius  discovered  the  variability  of  o  Ceti. 

1603.  Bayer's  Maps  of  the  Stars  published. 

1604.  Kepler  obtained  an  approximate  value  of  the  correction 
for  refraction. 

1608.  Jansen  and  Lippersheim,  of  Holland,  are  said  to  have 
invented  the  refracting  telescope,  using  a  convex  lens.     It 
is,  however,  a  disputed  point  as  to  who  really  invented  the 
telescope.     The  use  of  the  lens  seems  to  have  been  known 
about  fifty  years  before  this. 

1609.  Kepler  announced  his  first  two  laws. 

1610.  Galileo,  using  a  telescope  with  a  concave  object-lens,  dis- 
covered the  satellites  of  Jupiter,  the  librations  of  the  moon, 
the  phases  of  Venus,  and  some  of  the  nebulae. 

1611.  Spots  and  rotation  of  the  sun  discovered  by  Fabricius. 
1614.  Napier  invented  logarithms. 

1618.  Kepler  announced  his  third  law. 

.  The  first  recorded  transit  of  Mercury,  observed  by  Gas- 
sendi.     The  vernier  invented. 


252  APPENDIX. 

A.D. 

1633.  Galileo  forced  to  deny  the  Copernican  theory. 

1639.  First  recorded  transit  of  Venus,  observed  by  Horrox  aii«l 
Crabtree. 

1640.  Gascoigne  applied  the  micrometer  to  the  telescope. 
1646.  Fontana  observed  the  belts  of  Jupiter. 

1654.  The  discovery  of  Saturn's  rings  by  Huyghens.  (Galileo 
had  previously  noticed  something  peculiar  in  the  planet's 
appearance.) 

1658.  Huyghens  made  the  first  pendulum-clock.  (The  discovery 
is  also  ascribed  to  Galileo  the  younger.) 

1662.  Royal  Society  of  London  founded. 

1 663.  Gregory  invented  the  Gregorian  reflecting  telescope. 

1664.  Hook  detected  Jupiter's  rotation. 

1666.  Foundation  of  the  Academy  of  Sciences  at  Paris. 
1669.  Newton  invented  the  Newtonian  reflecting  telescope. 

1673.  Flamsteed  explained  the  equation  of  time. 

1674.  Spring  pocket-watches  invented  by  Huyghens.     (Said  also 
to  have  been  invented,  somewhat  earlier,  by  Dr.  Hooke.) 

1675.  Transmission  of  light  discovered  by  Romer.     Transit  In- 
strument used  to  determine  right   ascensions  by  Romer. 
Greenwich  Observatory  founded. 

1687.  Newton's  Principia  published. 

1690.  Ellipticity  of  the  earth  theoretically  determined  by  Huy- 
ghens. 

1704.  Meridian  Circle  used  by  Romer. 

1711.  Foundation  of  the  Royal  Observatory  at  Berlin. 

1725.  Compensation  pendulum  announced  by  Harrison. 

1726.  Mercurial  pendulum  invented  by  Graham. 

1727.  Aberration  of  light  discovered  by  Bradley. 
1731.  Hadley's  Quadrant  invented. 

1744.  Euler's  Theoria  Motuum  published,  the  first  analytical 
work  on  the  planetary  motions. 

1745.  Nutation  of  the  earth's  axis  discovered  by  Bradley. 
1750.  Wright's  Theory  of  the  Universe  published. 

1765.  Harrison  rewarded  by  Parliament  for  the  invention  of  the 

Chronometer. 

1767.  British  Nautical  Almanac  commenced. 
1769.  Transit  of  Venus  successfully  observed. 


ASTRONOMICAL   CHRONOLOGY.  253 

A.D 

1774.  Experiments  by  Maskelyne  to  determine  the  earth's  den- 
sity. 

1781.  Uranus  discovered  by  Sir  William  Herschel.  Messier's 
catalogue  of  Nebulae  published. 

1783.  Herschel  suspected  the  motion  of  the  whole  solar  system. 

1784.  Researches  of  Laplace  on  the  stability  of  the  solar  system. 

1786.  Publication  of  Herschel's  catalogue  of  1000  nebulae. 

1787.  Herschel  began  to  observe  with  his  forty-feet  reflector.  The 
Trigonometrical  Survey,  of  England  commenced. 

1788.  Publication  of  La  Grange's  Mecanique  Analytique. 

1789.  The  rotation  of  Saturn   determined  by  Herschel,  and  a 
second  catalogue  of  1000  nebulae  published. 

1792.  Commencement  of  the  Trigonometrical  Survey  of  France. 
1796.  French  Institute  of  Science  founded. 
1799.  Laplace's    Mecanique  Celeste  commenced. 
1801-7.  The  minor  planets  Ceres,  Pallas,  Juno,  and  Vesta  dis- 
covered. 

1802.  Publication  of  Herschel's  third  catalogue  of  Nebulae. 

1803.  Publication  of  Herschel's  discovery  of  Binary  Stars. 

1804.  Proper  motions  of  300  stars  published  by  Piazzi. 

1805.  Commencement  of  attempts  to  determine  the  parallax  of 
the  stars. 

1812.  Troughton's  Mural  Circle  erected  at  Greenwich. 

1820.  Foundation  of  the  Royal  Astronomical  Society  of  London. 

1821.  Observatory  at  Cape  of  Good  Hope  founded. 
1823.  Cambridge  (England)  Observatory  founded. 
1835.  Airy  determined  the  time  of  Jupiter's  rotation. 

1837.  Value  of  the  moon's  equatorial  parallax  determined  by 
Henderson.     The  East  Indian  arc  of  21°  21'  completed. 

1838.  Parallax  of  61  Cygni  determined  by  Bessel. 

1839.  Parallax  of  a  Centauri  determined  by  Henderson.     Im- 
perial Observatory  at  Pulkowa  (Russia)  founded. 

1840.  Harvard  Observatory  founded. 

1842.  Washington  Observatory  founded.     Mean  density  of  the 
earth  determined  by  Baily. 

1843.  Periodicity  of  the  solar  spots  detected  by  Schwabe. 

1844.  Taylor's  catalogue^of  11,015  stars.     Transmission  of  time 
by  electric  signals  commenced  in  the  United  States. 


254  APPENDIX. 

A.D. 

1845.  Discovery  of  the  fifth  minor  planet,  Astrsea.     (187  othera 
have  since  been  discovered.) 

1846.  Discovery  of  the  planet  Neptune. 

1847.  Lalande's  catalogue  of  47,390  stars  republished  by  the 
British  Association. 

1851.  Foucault's  pendulum  experiment  to  demonstrate  the  earth 'a 
rotation.     Completion  of  the  Russo-Soandinavian  arc. 

1854.  Difference  of  longitude  of  Greenwich  and  Paris  deter- 
mined by  electric  signals. 

1855.  Commencement  of  the  American  Ephemeris. 

1858.  De  La  Rue  obtained  a  stereoscopic  view  of  the  moon. 
(The  first  photograph  of  the  moon  was  made  by  Dr.  J.  W. 
Draper,  of  New  York,  in  1840.) 

1859.  Publication  of  Section  I.  of  Argelander's  "  Zones",  con- 
taining 110,982  stars.     Suspected  discovery  of  the  planet 
Vulcan,  lying  within  the  orbit  of  Mercury.    Completion  of 
the  Berlin  Star  Charts  commenced  in  1830. 

1861.  Appearance,  in  June,  of  a  comet  with  a  tail  of  105°  (the 
longest  on  record).     Publication  of  Section  II.  of  Argelan- 
der's "  Zones",  containing  105,075  stars. 

1862.  Section  III  of  Argelander's  "Zones",  containing  108,129 
stars.    Notes  on  989  Nebulae,  by  the  Earl  of  Rosse.    Bond's 
Memoir  on  Donati's  Comet  of  1858,  published  in  the  Annals 
of  the  Harvard  Observatory. 

1863.  Announcement  by  several  computers  that  the  received 
value  of  the  sun's  parallax  is  too  small   by  about  0".3. 
Spectroscopic  observations  of  celestial  objects,  by  Huggins 
and  Miller. 

1864.  Publication  of  Sir  John  Herschel's  great  catalogue  of 
5079  nebulae. 

1806.  Magnificent  display  of  shooting-stars  on  the  morning  of 
November  14th. 

.NOTE.  —  The  third  edition  of  Chambers' s  Descriptive  Astronomy  (Oxford, 
J877)  contains  an  exceedingly  interesting  and  elaborate  summary  of  As- 
tronomical Chronology. 


.NAVIGATION.  255 


NAVIGATION. 


THE  earliest  accounts  of  Navigation  appear  to  be  those  of 
Phoenicia,  1500  B.C.  Long  voyages  are  mentioned  in  the  earliest 
mythical  stories ;  but  the  first  considerable  voyage  of  even  pro- 
bable authenticity  seems  to  be  that  of  the  Phoenicians  about 
Africa,  600  B.C.  The  Roman  navy  dates  from  311  B.C.,  and 
that  of  the  Greeks  from  a  much  earlier  time.  After  the  decay 
of  these  nations,  commerce  passed  into  the  hands  of  Genoa, 
Venice,  and  the  Hanse  towns ;  from  them  it  passed  to  the  Portu- 
guese and  the  Spanish ;  and  from  them  again  to  the  English  and 
the  Dutch. 

The  attractive  po^ser  of  the  magnet  has  been  known  from 
time  immemorial ;  but  its  property  of  pointing  to  the  north  was 
probably  discovered  by  Roger  Bacon  in  the  thirteenth  century. 
When  first  used  as  a  compass,  the  needle  was  placed  upon  two 
bits  of  wood,  which  floated  in  a  basin  of  water ;  and  the  method 
of  suspending  it  dates  from  1302.  The  variation  of  the  compass 
was  discovered  by  Columbus,  in  1492.  The  compass-box  and 
the  hanging-compass  were  invented  by  the  Rev.  William  Bar- 
lowe,  in  1608. 

Plane  charts  were  used  about  1420.  The  discovery  that  the 
loxodromic  curve  is  a  spiral  was  made  by  Nonius,  a  Portuguese, 
in  1537.  The  log  is  first  mentioned  by  Bourne,  in  1577.  Loga- 
rithmic tables  were  applied  to  Navigation  by  Gunter,  in  1620; 
and  middle-latitude  sailing  was  introduced  three  years  after- 
wards. Other  dates  relating  to  the  subject  of  Navigation  are 
given  in  the  Astronomical  Chronology. 


256 


APPENDIX. 


TABLE  I. 

THE  PLANETS,  THE  SUN,  AND  THE  MOON. 


Inclina- 

Eccentri- 

Greatest 

Least  dis- 

Mean dis- 

Name. 

Symbol. 

the 

city 

of 

dista 

nee 

tance  f 

rom 

tance 

from 

Mean  distance 

Ecliptic. 

Orbit. 

from  Sun. 

Sun. 

Su 

a. 

from  Suu  in  miles. 

6    t 

! 

Mercury 

§ 

70008 

0.20560 

0.46669 

0.30750 

0.38710 

35,760,000 

Venus  .. 

9 

3  23  31 

0.00684 

0.72826 

0.71840 

0.72333 

66,820,000 

Earth  ... 

rjy  or  Q 

0.01679 

1.01679 

0.98321 

1.00000 

92,380,000 

Mars  

rf 

1  51  05 

0.09326 

1.66578 

1.38160 

1.52369 

140,800,000 

Jupiter. 

"4 

1184C 

0.04824 

5.45378 

4.95182 

5.20280 

480,600,000 

Saturn... 

h 

2292S 

0.05560 

10.07328 

9.00442 

9.53885 

881,200,000 

Uranus.. 

(?)  or  $ 

0  46  3C 

0.04658 

20.07612 

18.28916 

19.18264 

1,772,000,000 

Neptune 

W 

1465S 

0.00872 

30.29888 

29.77506 

30.03697 

2,775,000,000 

Moon.... 

£ 

5084C 

0.05491 

1 

Name. 

Sidereal  Period 
in  days. 

DailyHe- 

liocentric 
Motion. 

Synodic 
Period 
in  days. 

Max.  Diam- 
eter from  ©. 

Min.  Diam- 
eter from  © 

Diam.  at 
mean 
distance. 

Diameter 
in  miles. 

o    t      n 

tt 

ff 

tt 

Mercury 

87.969 

40533 

115 

877 

12.9 

04-  ^ 

06.7 

3,000 

Venus... 

224.701 

13608 

583.921 

1'  06.3 

09'.7 

17.1 

7,600 

Earth  ... 

365.256 

5908 

7,925.6 

Mars  

686.980 

3127 

779 

936 

30.1 

04.1 

11.1 

4,100 

Jupiter. 

4,332.585 

459 

398 

884 

50.6 

30.S 

37.2 

89,000 

Saturn... 

10,759.220 

201 

378 

092 

20.3 

16.1 

72,000 

Uranus.. 

30,686.821 

42 

369.656 

04.3 

O3.c 

03.9 

33,000 

Neptune 

60,126.722 

22 

367 

488 

02.9 

02.6 

02.7 

37,000 

Moon.... 

27.322 

29.531 

33'  31 

28'  48 

31  '07 

2,161.6 

Sun  

32    35.6 

31 

31 

32  03.6 

861,400 

Volume. 

Mass. 

Density. 

Rotation. 

Inclination 

App.  Diam. 

Name. 

of  A> 

tis  to 

of  Sun 

Ecliptic. 

fromPlanet. 

d.    h 

m. 

a. 

t       tt 

Mercury 

0.052 

•Jff¥n7J(TTT 

2.02 

24  05 

30 

82  49 

Venus... 

0.851 

^oo'c 

"Fff 

0.90 

23  21 

23 

73° 

32' 

44  19 

Earth  ... 

1.000 

32Ti 

1 

00 

23  56 

04 

66 

33 

32  04 

Mars  

0.239 

0.50 

24  37 

23 

61 

18 

21  02 

Jupiter  . 

1,387.431 

ToX 

f¥ 

0.23 

< 

)  55 

21 

86 

55 

6  10 

Saturn... 

746.898 

ri 

J-Q 

0.13 

10  14 

24 

62 

3  22 

Uranus.. 

72.359 

Tn 

OU 

0 

18 

1  40  1 

Neptune 

98.664 

OfJ 

0 

18 

1  04 

Moon.... 

0.020 

2T7J  0  fj  TToTT 

0.63 

27  07  43 

11 

88 

30 

Sun  

1,284,000 

0-25 

25  07  48 

82 

30 

TABLES.  257 

Too  much  confidence  must  not  be  placed  in  the  absolute  accuracy  of  all 
the  elements  above  tabulated.  The  necessity  of  this  caution  will  become 
obvious  to  any  one  who  will  compare  the  values  of  these  elements  as  they 
are  given  by  different  authorities.  The  relative  distances,  the  apparent 
diameters,  and  the  periods  of  the  planets,  being  matters  of  direct  observa- 
tion, are  known  to  a  great  degree  of  accuracy ;  but  the  absolute  distances 
and  diameters,  and  the  volumes,  depending  as  they  do  upon  the  distance 
of  the  earth  from  the  sun,  must  be  considered  to  be  only  approximately 
known.  The  masses,  too,  of  some  of  the  planets  are  uncertain :  and  so  also 
must  be  the  densities,  which  depend  upon  the  masses. 

TABLE  II. 

THE   EARTH. 


~ 


Density 5.67  (water  being  1) 

Polar  Diameter 7899.170  miles. 

Equatorial  "     7925.648 

Compression 0.00334 

Length  of  sidereal  year 365d.  06h.  09m.  09.6s. 

"  tropical     "     "     05     48       47.8 

"         "anomalistic" "     06     13      49.3 

"         "sidereal  day 23    56      04.09 

Eccentricity  of  orbit 0.0167917 

Inclination     "      "     in  1868 23°  27' 23^' 

Annual  diminution 0".4645 

"       precession 50".24 

"       advance  of  line  of  nodes 11". 8 


TABLE  III. 

THE   MOON. 


Distance  from  earth  in  earth's  radii 60.267 

Mean  distance  from  earth 238,800  miles. 

Greatest    "         "         "     257,900       " 

Least         "         "        " 221,400      " 

Sidereal  revolution 27d.  07h.  43m.  11.5s. 

Synodical         "        29     12     44      03 

Mean  daily  geocentric  motion 13°  10'  36" 

Eevolution  of  nodes 6793.43  days. 

"  perigee 3232.58     " 

Mean  horizontal  parallax 57'  03" 

Greatest        "  "        61    32 

Least  "  "       52   50 

Radius  in  terms  of  earth's  radius 0.2727 

Mass       "      "       "       "        mass j\ 

Density"      "       "       "        density f 


253 


APPENDIX. 


TABLE  IV. 

SATELLITES. 
SATELLITES   OF   MARS. 


Names. 

Inclination  of 
orbit  to  orbit  of 
Primary. 

Instance  from 
Primary  in 
miles. 

Sid.  Period. 

Diameter  in 
miles. 

Magni- 
tude. 

I.  Phoblis.... 
II.  Deimos  

[They  move  very 
iifHriyin  the  plane 
oi'Mars'sequator.] 

5,800 
14,500 

d.     h.     m. 
0   07    39 

1  06  18 

[Not  yet  de- 
termined, but 
very  small.] 

11 

12 

SATELLITES   OF   JUPITER. 

I    Jo  

O     1       tt 

3  05  30 
3  04  25 

3  00  28 

2  58  48 

208,000 
426.000 
($79,000 
1,200,000 

1  18  28 
3  13  15 
7  03  43 
16  16  32 

2400 
2100 
3500 
3000 

7 
7 
6 

7 

II.  Enropa.  •  -. 
III.  Ganvinede 
IV.  Callisto.... 

SATELLITES    OF    SATURN. 

I.  Mimas  
II.  Knee  lad  us 
III  Tethvs..  . 

[The  first  7  move 

122.000 
156,000 
193,000 
248,000 
347,000 
805,000 
1,018,000 
2,334,000 

0  22  37 

1  08  53 
1  21  18 
2  17  41 
4  12  25 
15  22  41 
21   07  07 
79  07  54 

1000 

500 
500 
1200 
3300 

1800 

17 
15 
13 

12 
10 
8 
17 
9 

IV.  Dion'e  
V.  Ehea  
VI.  Titan  
VII.  Hyperion. 
VIII.  lapetii.s.... 

plane  of  tlie  plan- 
et's equator  :    the 
8th  in  a  pl.-me  in- 
clined   about    120 
to  that  plane  J 

SATELLITES   OF   URANUS. 

I.  Ariel  
Il.Umbriel... 
III.  Titania  
IV.Oberon  

[The  motion  of 
the     satellite.-,     is 
retrograde,    in    a 
plane  inclined  790 
to  the  plane  of  the 
ecliptic.] 

124,000 
173,000 
284.000 
380,000 

2  12  48 
4  03  27 
8  16  56 
13  11  07 

SATELLITE   OF   NEPTUNE. 

I. 

290  to   ecliptic; 
motion  retrograde 

220,000 

5  21  08 

14 

TABLES. 


259 


TABLE  V. 

THE   MINOR    PLANETS. 


No. 

Name. 

Year 
of  Dis- 
cov- 
ery. 

Discoverer. 

Inclina- 
tion. 

Eccen- 
tricity. 

Period 
Years. 

Distance 
in  Radii 
of  Earth's 
orbit. 

Diam- 
eter in 
miles. 

1 

Ceres 

1801 

Piazzi    .... 

10°  36' 

0  080 

46 

2  77 

227 

9 

Pallas 

1802 

Olbers  

34   42 

0  240 

46 

2  77 

172 

3 

Juno  

1804 

Hard  in0"  

13   03 

0256 

44 

2  67 

112 

4 

Vesta 

1807 

Olbers 

7   08 

0  090 

36 

2  36 

228 

5 
6 

Astrsea  
Hebe  

1845 
1847 

Hencke  
Hencke  

5   19 
14  46 

0.190 
0201 

4.1 

3  8 

2.58 
2  43 

61 

100 

7 
8 

Iris  
Flora 

Hind  
Hind  

5   27 
5  53 

0.231 
0  157 

3.7 
33 

2.39 
2  20 

96 
60 

q 

Metis  

1848 

Graham  . 

5   36 

0  123 

37 

2  39 

76 

10 
11 
12 
13 

Hygeia  
Parthenope 
Victoria.... 
Egeria  

1849 
1850 

De  Gasparis.  . 
De  Gasparis  . 
Hind  

De  Gasparis.' 

3  47 
4   36 
8   23 
16   32 

0.101 
0.099 
0.219 
•0  088 

5.6 

3.8 
5.6 
4  1 

3.15 
2.45 
2.33 

2  58 

111 
62 
41 
73 

14 

Irene  

1851 

Hind  

9   07 

0  165 

42 

259 

68 

15 

Eunomia... 

De  Gasparis.  • 

11   44 

0  188 

43 

264 

12 

16 
17 

Psyche  
Thetis  

1852 

De  Gasparis.. 
Luther  

3   04 
5   35 

0.136 
0  127 

5.0 
3  9 

2.93 
247 

93 
52 

18 
10 

Melpomene 



Hind  
Hind 

10  09 
1   32 

0.217 
0  158 

3.5 

3  8 

2.30 
2  44 

54 
61 

20 
21 

Massilia.... 
Lutetia  ... 

De  Gasparis.  . 
Goldschmidt. 

0  41 
3   05 

0.144 
0  162 

3.7 
3  1 

2.41 
244 

68 
40 

22 

Calliope.  ••• 

Hind  

13   44 

0  104 

50 

291 

96 

23 
24 
25 

Thalia  
Themis  
Phocea  • 

1853 

Hind  
De  Gasparis. 
Chacornac.  •• 

10   13 
0  48 
21   34 

0.232 
0.117 
0  253 

4.3 

5.6 
37 

2.62 
3.14 
240 

42 
36 
31 

26 

Proserpine 

Luther  

3   35 

0088 

43 

266 

47 

27 
?8 

Euterpe  .... 
Bellona 

1854 

Hind  

Luther        .  . 

1   35 

9   21 

0.173 
0  150 

3.6 
46 

2.35 

2  78 

39 
59 

29 
30 
31 

Amphitrite 
Urania  
Euphrosyne 

Marth  
Hind  
Ferguson  

6   07 
2  05 
26   25 

0.072 
0.127 
0216 

4.1 
3.6 
5.6 

2.55 
2.36 
3  16 

83 
51 
50 

32 
33 
34 

Pomona  .... 
Polyhymnia 
Circe  

1855 

Goldschmidt. 
Chacornac.  .. 
Chacornac..  • 

5   29 
1   56 
5   26 

0.082 
0.338 
0110 

4.2 
4.8 
44 

2.58 
2.86 
268 

35 

38 
29 

So 

'•8   10 

0  214 

5  2 

301 

25 

36 

37 

Atalanta  ... 
Fides  



Goldschmidt. 
Luther  

18   42 
3   07 

0.298 
0  175 

4.6 

43 

2.75 

264 

20 
41 

38 

Leda  

1856 

Chacornac  .... 

6   58 

0  156 

45 

274 

29 

30 

Ltetitia 

10  21 

0  111 

46 

277 

87 

40 
41 
49 

Harmonia. 
Daphne  .... 
Isi<s 



Luther  
Goldschmidt. 

4   15 
16  45 
8   35 

0.046 
0.270 
0226 

3.4 
4.6 
38 

2.27 

2.77 
244 

43 

1857 

Pogson  .  . 

3  27 

0  168 

33 

220 

44 

Goldschmidt. 

3  41 

0  149 

38 

242 

45 

46 

47 

Eugenia.... 
Hestia  
Melete 



Goldschmidt. 
Pogson  
Goldachmidt 

6   34 
2   17 
8   01 

0.082 
0.162 
0  237 

4.5 
4.0 
42 

2.72 
2.52 
2  60 

L  

260 


APPENDIX. 


TABLE  V.—  Continued. 


No. 

48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 

8 

78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 

Name. 

Year 
of  Dis- 
cov- 
ery. 

Discoverer. 

Inclina- 
tion. 

Eccen- 
tricity. 

Period 
in 

Years. 

Distance 
in  Radii 
of  Earth's 
orbit. 

Diam- 
eter in 
Miles. 

Aglaia  •  ... 

1857 

5°  00' 
6   29 
3   08 
2  47 
10   14 
7   24 
5   07 
11   47 
7   20 
15  04 
5   02 
18   17 
8   36 
2   12 
3   34 
5  45 
1    19 
3   28 
3  04 
5   59 
8   28 
7   58 
11    39 
5   24 
23   19 
2   25 
3   59 
5   00 
2   02 
2   28 
8   39 
4  37 
8   37 
7   56 
2   51 
5  02 
9   22 
11   56 
4  48 

5  09 
15   13 

0.128 
0.076 
0.238 
0.287 
0.063 
0.004 
0.213 
0.199 
0.139 
0.108 
0.041 
0.163 
0.119 
0.170 
0.185 
0.127 
0.125 
0.120 
0.154 
0.184 
0.175 
0.186 
0.183 
0.120 
0.174 
0.044 
0.238 
0.307 
0.188 
0.136 
0.205 
0.195 
0.200 
0.212 
0.226 
0.084 
0.238 
0.194 
0.205 
0.081 
0.167 
0.205 
0.148 

4.9 
5.5 
5.4 
4.3 
3.7 
5.5 
4.2 
4.6 
4.6 
5.6 
4.4 
5.1 
4.5 
5.5 
3.7 
3.7 
4.4 
6.7 
4.3 
3.8 
5.2 
4.6 
4.2 
3.4 
4.6 
4.3 
4.6 
4.4 
6.2 
4.4 
4.2 
3.8 
3.5 
4.S 
4.6 
3.8 
3.6 
4.3 
5.4 
6.5 
4.6 
4.0 
5.5 

2.88 
3.10 
3.09 
2.65 
2.38 
3.10 
2.61 
2.71 
2.77 
3.16 
2.70 
2.97 
2.71 
3.13 
2.39 
2.40 
2.68 
3.42 
2.65 
2.42 
2.99 
2.77 
2.61 
2.27 
2.76 
2.66 
2.78 
2.67 
3.39 
2.67 
2.62 
2.44 
2.30 
2.86 
2.76 
2.43 
2.37 
2.66 
3.09 
3.49 
2.75 
2.53 
31.2 

Goldschmidt. 
Goldschmidt. 
Ferguson  
Laurent  
Goldschmidt. 
Luther  

Pales  

Virginia.... 
Nemausa.... 

1858 

Calypso  

Alexandra.. 
Pandora  . 



Goldschmidt. 
Searle 

Mnemosyne 
Concordia... 
Daniie  

1859 
1860 

Luther 

Luther  

Goldschmidt. 
Chacornac  

Olympia.... 
Erato  



Echo  

Ferguson  
De  Gasparis.. 
Tern  pel  

Ausonia  
Angelina.... 
Cybele  

1861 

Maia  

T  uttle 

Pogson  

Hesperia.... 
Leto  



Schiaparelli.. 

Panopea  .  .. 

Goldschmidt. 
C.H.F.  Peters 

Luther  .  . 

Feronia  

Niobe  

Clyde  

1862 

Tuttle  

Galatea..  .. 

Tempel  
Peters 

Eurvdice.... 
Preia  

D'  Arrest  
Peters  

Friar  o-a  



Diana  

1863 
i'864 

Luther  

Eurynome.. 
Sappho  
Terpsichore 
Alcmene.... 
Beatrix  .... 

Watson  

Pogson 

Xempel.... 

1865 

Luther  

De  Gasparis.  . 
Luther 

Clio  

Io  

Peters  

Seraele  . 

1866 

Tietjen 

Svlvia  

Thisbe  

Peters  
Stephan  
Luther  
Stephan  

Julia  .« 

Antiope 

jEsfina.  .. 

The  number  of  minor  planets  discovered  up  to  October,  1878,  was  192. 
The  diameters  are  derived  from  photometric  experiments  made  by  Pro- 
fessor Stampfer,  of  Vienna.  They  are  probably  only  relatively  correct. 


TABLES. 


261 


TABLE  VI. 

SCHWABE'S  OBSERVATIONS  OF  THE  SOLAR  SPOTS, 


Ysar. 

Number 
of  days 
of  obser- 
vation. 

New  Groups. 

Days  on 
which  the 
Sun  was  free 
from  spots. 

Year. 

Number 
of  days 
of  obser- 
vation. 

New  Groups. 

Days  on 
which  the 
Sun  was  free 
from  spots. 

1826 

277 

118 

22 

1846 

314 

157 

1 

7 

273 

161 

2 

7 

276 

257 

0 

8 

282 

225(Max.) 

0 

8 

278 

330(Max.) 

0 

9 

244 

199 

0 

9 

285 

238 

0 

1830 

217 

190 

1 

1850 

308 

186 

2 

1 

239 

149 

3 

1 

308 

151 

0 

2 

270 

84 

49 

2 

337 

125 

2 

3 

247 

33(Min.) 

139 

3 

299 

91 

3 

4 

273 

61 

120 

4 

334 

67 

65 

5 

244 

173 

18 

5 

313 

79 

146 

6 

200 

272 

0 

6 

321 

34(Min.) 

193 

7 

168 

333(Max.) 

0 

7 

324 

98 

52 

8 

202 

282 

0 

8 

335 

188 

0 

9 

205 

162 

0 

9 

343 

205 

0 

1840 

263 

152 

3 

1860 

332 

211(Max.) 

0 

1 

283 

102 

15 

1 

322 

204 

0 

2 

307 

68 

64 

2 

317 

160 

3 

3 

312 

34(Min.) 

149 

3 

330 

124 

2 

4 

321 

52 

111 

4 

325 

130 

4 

1845 

332 

114 

29 

1865 

307 

93 

25 

TABLE  VII. 

PERIODIC    COMETS. 


Name. 

Last  perihelion 
passage  observed. 

Longi- 
tude of 
Asc.Node 

Inclina- 
tion. 

Eccentri- 
city. 

Semi- 
major 
axis. 

Period 
in 
Days. 

Motion. 

Encke's  
Winnecke's... 
Brorsen's  

1878  
1875,  Mar.  12 
1879,  Oct.  12 
1  Sf>i>,  Sept.  23 
1877  
1873,  JnlvlS 
1871,  Dec.    1 
1835,  Nov.  16 

o        t 

334  31 
113  31 
101  46 
245  57 

148  27 
209  40 
268  54 
55  10 

0        1 

13  05 
10  48 
29  45 
12  34 
13  56 
11  22 
54  32 
17  45 

0.847 
0.755 
0.802 
0.756 
0.660 
0.558 
0.830 
0.967 

2.22 
3.14 
3.14 
3.50 
3.44 
3.81 
6.03 
17.99 

1210 
2020 

2037 
2415 
2366 
2715 
4986 
28000 

Direct. 
Direct. 
Direct. 
Direct. 
Direct. 
Direct. 
Direct. 
Ketro. 

D'  Arrest's.... 
leave's  

Mechain's  
Halley's  

262 


APPENDIX. 


TABLE  VIII. 

TRANSITS   OF   THE   INFERIOR   PLANETS. 


Mercury. 

Venus. 

1802. 

November    8. 

1639. 

December    4. 

1815. 

November  11. 

1761. 

June            5. 

1822. 

November    4. 

1769. 

June            3. 

1832. 

May              5. 

1874. 

December    8. 

1835. 

November    7. 

1882. 

December    6. 

1845. 

May               8. 

2004. 

June            7. 

1848. 

November    9. 

2012. 

June             5. 

1861. 

November  11. 

2117. 

December  10. 

1868. 

November    4. 

2125. 

December    8. 

1878. 

May              6. 

2247. 

June           11. 

1881. 

November    7. 

2255. 

June            8. 

1891. 

May              9. 

2360. 

December  12. 

1894. 

November  10. 

2368. 

December  10. 

TABLE  IX. 

LIST   OF   STARS   WHOSE   ANNUAL   PARALLAX   HAS   BEEN 

COMPUTED. 
(All  these  are  doubtful.) 


Stars. 

Parallax. 

Distance  in  radii 
of  the  Earth's 
orbit. 

Observer. 

o.  Centauri  

0.92 

224,000 

Maclear.  Henderson 

61  Cygni 

/  0.35 

589,000 

Bessel. 

21258  Lalande  
17415  Oeltzen  

(.0.56 
0.27 
025 

368,000 
764,000 
825  000 

Auvers. 
Auvers. 
Kriiger. 

1830  Groombridge.. 
70  Ophiuchi  
a  Lyne  

0.23 
0.16 
0  155 

897,000 
1,289,000 
1  331  000 

C.  A.  Peters. 
Kr  tiger. 
W.  &  O.  Struve. 

Sirius  

0150* 

1  375,000 

Henderson.     Peters. 

i  Ursse  Majoris.. 
A  returns  
Polaris    

0.133 
0.127 
007 

1,550,000 
1,624,000 
2  950,000 

C.  A.  Peters. 
C.  A.  Peters. 
C.  A.  Peters. 

Capella  

005 

4,130,000 

C.  A.  Peters. 

. 

Another  determination  gives  0".23. 


TABLES. 


263 


TABLE  X. 

THE   PRINCIPAL   CONSTELLATIONS. 
Those  found  in  Ptolemy's  Catalogue  (137  A.n.)  are  in  Italics. 
-    THE    NORTHERN    CONSTELLATIONS. 


No. 

Name. 

Coordinates  of  Centre. 

R.A.                    D. 

Name  of 
Principal  Star  of 
1st  or  2d  magnitude. 

Number 
of  stars  of 
1st  mag. 

Number  1 
of  stars 
of  first 
ftve  mag- 
nitudes. 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

29 

h.       m. 
1         0 
1      10 

2       0 
3    30 
5    45 
6       0 
7     55 
10    05 
10    40 
12    40 
13      0 
14    35 
15      0 
15    40 
15    40 
16    45 
17     20 
17     50 
18     10 
18     40 
19     30 
19    40 
20       0 
20     20 
20     20 
20     40 
21       0 
21     40 
22     25 

0 

35 

60 
32 
47 
68 
42 
50 
36 
58 
26 
40 
30 
78 
30 
10 
27 
66 
5 
15  S. 
35 
10 
18 
25 
42 
44 
15 
6 
65 
15 

Alpheratz  (a) 

Mirfak  (a) 
Capella  (a) 

Dubhe  (a) 

Cor  Caroli  (a) 
Arcturus  (a) 
Polaris  (a) 

Unnkalhay  (a) 
Rasalgeti  (a) 
Thuban  (a) 

Vega  (a) 
Altair  (a) 

Deneb  (a) 
Markab  (a) 

1 

1 
1 
1 

1 
1 

18 
46 
5 
40 
36 
35 
28 
15 
53 
20 
15 
35 
23 
19 
23 
65 
80 
6 
4 
18 
33 
5 
23 
67 
13 
10 
5 
44 
43 

Camelopa.rd.us  

Aurioci  •••• 

UTSCL  J\fctjor  

Coma  Berenicis  

Bootes     

Corona  Borealis  

Draco    

TaurusPoniatowskii 
Clypeus  Sobieskii... 

Vulpecula  et  Anser. 

Total 

6 

827 

264 


APPENDIX. 

TABLE  X.—  Continued. 

THE    ZODIACAL    CONSTELLATIONS. 


No. 
1 

2 
3 

4 
5 
6 
7 
8 
9 
10 
11 
12 

12 

Name. 

Coordinates  oi  Centre. 

K.A.                  P. 

Name  of 
Principal  Star  of 
1st  or  2d  magnitude. 

Number 
of  stars  of 
1st  laag. 

Number 
of  stars 
of  first 
five  mag- 
nitudes. 

Aries  

h.    m. 
2      30 

4      0 
7      0 

8    40 
10     20 
13    20 
15      0 
16     15 
18    55 
21       0 
22      0 
0    20 

18  N. 
18 

25 

20 
15 
3N. 

15  S. 
26 
32 
20 
9S. 
ION. 

Hamal  (a) 
Aldebaran  (a) 
f  Castor  (a) 
\  Pollux  (0) 

Regulus  (a) 
Spica  (a) 
Zubenelg  (a) 
An  tares  (a) 

SecundaGiedi(a) 
Sadalmelik  (a) 

1 
1 

1 
1 

1 

17 

•58 

28 

15 

47 
39 
23 
34 
38 
22 
25 
18 

Taurus  

Cancer  

Libra   

Scorpio  

Sagittarius  

JPisces  

Total  

5 

364 

THE   SOUTHERN   CONSTELLATIONS. 

1 
2 
3 

4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 

n 

18 
19 
20 
21 

Apparatus  Sculptoris. 

0    20 
1       0 
2      0 
2    20 
2    40 
3     15 
3    40 
4      0 
4    40 
4    40 
5    20 
5    25 
5     25 
5     25 
5     30 
6     45 
7       0 
7     25 
7     40 
7     40 
10      0 

32 
50 
12 
30 
70 
57 
30 
62 
62 
42 
75 
35 
55 
20 
0 
24 
2 
5 
50 
68 
0 

Menkar  (a) 
Achernar  (a) 

Phact  (a) 

Arneb  (a) 
Rigel  (ft) 
Sirius  (a) 

Procyon  (a) 
Canopus  (a) 

1 

2 
1 

1 
2 

13 
32 
32 
6 
25 
11 
64 
11 
17 
6 
9 
15 
17 
18 
37 
27 
12 
6 
133 
9 
3 

Qetus     

Fornax  Chemica  .  ... 
Hvdrus  

Horologiuni  

Reticulus  Rhomboidalis  .. 

Csjla  Sculptoris  

J^fons  Mensae  

Columba  Noachi  .  ... 
Equuleus  Pictoris... 
Lspus  

Orion  

Monoceros  .     ... 

Canis  Minor  

Piscis  Volaiis  
Sextans  

TABLES. 


265 


TABLE  X.—  Continued. 

THE   SOUTHERN   CONSTELLATIONS. 


No 

Name. 

Coordinates  of  Centre. 

R.A.                   D. 

Name  of 
Principal  Star  of 
1st  or  2d  magnitude. 

Number 
ofsta.rsof 
1st  mag. 

Number 
of  stars 
of  first 
five  mag- 
nitudes. 

22 
23 
24 
J25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
33 
39 
40 
41 
42 
43 
44 
45 

h.    m. 

10      0 
10      0 
10    50 
11     20 
12    15 
12    20 
12    25 
13      0 
15      0 
15    20 
15    25 
15    40 
16      0 
17      0 
17      0 
18    30 
18    40 
19    20 
20    40 
21      0 
21      0 
21     40 
22    20 
23    45 

0 

10 
35 
78 
15 
60 
18 
68 
48 
64 
76 
45 
65 
45 
54 
0 
40 
53 
68 
37 
55 
80 
32 
47 
66 

Alphard  (a) 
Algorab  (a) 

Rasalague  (a) 
Fomalhaut  (a) 

1 

2 
1 

49 
7 
17 
9 
10 
8 
7 
54 
2 
7 
34 
11 
12 
15 
40 
7 
8 
27 
5 
15 
16 
16 
11 
21 

Antlia  Pneumatica.. 
Cliamselcon  

Crater 

Crux    

Corvus  

Musca  Australia  

Circinus  -  •  

A  pus  

TriangulumAustrale 
Norma   

Ara  

Corona  Australls  
Tel6scopium  

Pavo  

Indus  ...      ... 

Octans  

Piscis  Australia  

Toucan  

45 

86 
1 

Total  

11 

911 

22 

2102 

23 


266 


APPENDIX. 


TABLE   XL 

VARIABLE   STARS. 


Name. 

Coordinates,  1870. 

R.A.                         D. 

Period  in 
Days. 

Change  of 
Magnitude. 

1 

Authontj. 

h.    m.    s. 

o       t 

a  Cassiopeise. 

0  33  09 

55  49.4  N 

79.1 

2  to  2.5 

Birt,                1831 

o  Ceti      ...    . 

2  12  47 

3  34.1  S 

331.34 

2  "  12 

D.  Fabricius  1596 

/?Persei  

2  59  43 

40  27.2  N 

2.87 

2.5"   4 

Montanari,      1669 

A  Tauri  

3  53  29 

12  07.3  N 

3.95 

4"    4.5 

Baxendell,      1848 

e  Aurigae  

4  52  38 

43  37.7  N 

250. 

3.5"   4.5 

Heis,               1846 

fj-  Doradus.... 

5  05  47 

61  58.4  S 

Long. 

5     "    9 

Moesta,           1865 

a  Orionis  

5  48  08 

7  22.8  N 

196. 

1     "    1.5 

J.  Herscliel,   1836 

C  Gerainorum 

6  56.24 

20  45.5  N 

10.16 

3.8"   4.5 

Schmidt,          1847 

a  Hydrse.  

9  21  12 

8  05.9  S 

55. 

2.5"    3 

J.  Herschel,   1837 

R  Leonis  

9  40  34 

12  01.  8  N 

331. 

5     "11.5 

Koch,              1782 

77  Argus  

10  40  02 

59    0.1  S 

46yrs. 

1     "    4 

Burchell,         1827 

R  Hydra?  

13  22  37 

22  36.4  S 

447.8 

4     "10 

Maraldi,          1704 

Z  Virginia.... 

13  27  45 

1232.7  8 

? 

5     "    8 

Schmidt,         1866 

T  Coronae  

15  54  04 

26  17.5  N 

? 

2.5"    9.8 

Bermingham,  1866 

30  Herculis... 

16  24  22 

42  10.1  N 

106. 

5     "    6 

Baxendell,      1857 

Nov.  Ophiu... 

16  52  13 

12  41.4  S 

? 

4.5  "  13.5 

Hind,              1848 

a  Herculis.... 

17  08  43 

14  32.4  N 

? 

3.1"    3.9 

W.  Herschel,  1795 

R  Clyp.  Sob. 

18  40  33 

5  50.5  S 

89. 

5     "    9 

Pigotl,             1795 

(3  Lyra?  

18  45  17 

33  12.7  N 

12.91 

3.5"   4.5 

Goodricke,      1784 

13  Lyra  

18  51  23 

43  46.6  N 

46. 

4.2"    4.6 

Baxendell,      1855 

^2  Cygni  

19  45  34 

32  35.2  N 

409.2 

5     "13 

Kirch,             1686 

T?  Aquilse  

19  45  51 

040!4N 

7.18 

3.6"    4.4 

Pigott,              1784 

34  Cygni  

20  13    0 

37  37.8  N 

18yrs. 

3     "    6 

Jansen,            1600 

RCephei  

20  23  41 

8844.   N 

73  yrs. 

5     "11 

Pogson,           1851. 

fj-  Cephei  

21  39  31 

58  11.1  N 

5  or  6yrs. 

4     "    6 

W.  Herschel,  178S 

6  Cephei  

22  24  21 

5745.   N 

5.37 

3.7"    4.8 

Goodricke,      1784 

j3  Pegasi  

22  57  28 

27  22.',  N 

? 

2     "    2.5 

Schmidt,         184.1 

1 

TABLES. 


267 


TABLE  XII. 

BINARY   STARS. 


Name. 

Co-ordinates,  1870. 

K.A.                        D. 

Magnitude 
of  Com- 
ponents. 

Semi- 
major 
Axis. 

Eccen- 
tricity. 

Period 
in 

Years. 

Calculator. 

£  Herculis.  ... 

h.    m.     a. 

16  36 

O        f 

3151    N 

3  —6 

n 
1.25 

0.45 

36.3 

Villarceau. 

rj  CoronaeBor. 

15  18 

3047    N 

6   —  6J 

0.95 

0.28 

43.6 

Winnecke. 

f  Cancri  

8  04  45 

18  02.4  N 

6-7-7£ 

1.29 

0.23 

58.9 

Madler. 

f  UrsaeMaj... 

11  11  15 

3216    N 

4  -6J 

2.45 

0.39 

63.1 

J.  Breen. 

a  Centauri.... 

14  30  48 

60  17.9  S 

1   —2 

15.50 

0.95 

80.0 

E.B.Powell. 

w  Leonis  ...... 

9  21 

939    N 

6£-7£ 

0.85 

0.64 

82.5 

Madler. 

r  Ophiuchi... 

17  56 

811     S 

5  —6 

0.82 

0.037 

87. 

Madler. 

70  Ophiuchi.. 

17  58  52 

2  22.5  N 

4£  —  7 

4.19 

0.44 

92.8 

Madler. 

A  Ophiuchi... 

16  24 

217    N 

4  —6 

0.84 

0.477 

95. 

Hind. 

7  Coronas  Aus. 
I  Bootis  

18  57  38 
14  45  22 
19  41 

37  14.7  S 
19  38.5  N 
4448    N 

6   —6 
3i  —  9 

2.54 
12.56 
1.81 

0.60 
0.59 
0.60 

100.8 
117.1 

178.7 

Jacob. 
J.Herschel. 
Hind. 

6  Cygni  

ri  Cassiopeiae  . 

0  41  09 

57  07.8  N 

4  -7* 

10.33 

0.77 

181. 

E.B.Powell. 

7  Virginia...  . 

12  35  05 

044.2S 

4   —4 

3.58 

0.87 

182.1 

J.Herschel. 

o  CoronseBor. 

16  10 
7  26  18 
21  01  04 

3412    N 
32  10.4  N 
38  05.1  N 

6  —  6J 
3   —  3J 
5£  —  6 

2.71 
8.08 
15.4 

0.30 
0.75 

195.1 
252.6 
452. 

Jacob. 
J.Herschel. 

61  Cygni  

fi  Bootis  

15  19  35 

3750    N 

4  —8 

3.21 

0.84 

649.7 

Hind. 

y  Leonis  

10  12  47 

20  30.1  N 

2  —4 

1200. 

INDEX. 


[The  references  are  to  the  pages.] 


Aberration  of  light,  116;  diurnal  and 
annual,  117;  separated  from  par- 
allax, 221. 

Acceleration,  secular  of  the  moon,  133. 

Aerolites,  209. 

Algol,  224. 

Altazimuth,  39. 

Altitude,  19. 

Altitude  and  azimuth  instrument,  38; 
use  of,  40. 

Altitudes,  method  of  equal,  39. 

Amplitude,  19. 

Andromeda,  nebula  in,  231. 

Annular  eclipse,  141. 

Anomaly,  90. 

Aphelion,  90. 

Apogee,  122. 

Appulse,  136. 

Apsides,  of  earth,  116;  of  moon,  122; 
of  planets,  164. 

Arc  of  meridian  measured,  59. 

Argo,  nebula  in,  234. 

Aries,  first  point  of,  20,  84. 

Ascension,  right,  20 ;  related  to  sidereal 
time,  22. 

Asteroids,  171;  table  of,  260. 

Astronomy,  11;  chronology  of,  248. 

Atmosphere,  height  of,  52. 

Attraction,  law  of,  108. 

Axis,  of  the  heavens,  15;  of  the  earth, 
17;  of  rotation  and  collimation,  32. 

Azimuth,  19. 

Base-line,  60. 
Bode's  law,  171. 
Branches  of  meridian,  18. 

Calendar,  105. 

Centauri,  alpha,  distance  of,  222;   a 

binary  star,  227. 
Centrifugal  force,  65,  247. 
Ceres,  discovery  of,  172. 


Chronograph,  29. 

Chronometer,  Greenwich  time  given 
by,  77. 

Circle,  vertical,  19 ;  hour,  20 ;  of  per- 
petual apparition,  20;  diurnal,  24; 
of  latitude,  84. 

Circle,  meridian,  34;  mural,  38;  re- 
flecting, 48. 

Clock,  astronomical,  28;  driving,  41. 

Clusters  of  stars,  229. 

Coal  sack,  234. 

Collimation,  axis  of,  32. 

Colures,  84. 

Comets,  187;  diversity  of  appearance, 
188;  tail,  1 89 ;  orbits,  191 ;  periods  and 
motion,  192 ;  mass  and  density,  193  ; 
light,  194;  periodic,  195;  Encke's, 
195 ;  Winnecke's  or Pons's,197 ;  Bror- 
sen's,  197;  Biela's,  197;  D'Arrest's, 
198;  Faye's,  198;  Mechain's,  199; 
Halley 's,  199 ;  Great,  of  1 811 , 200 ;  of 
1843,  200  ;  Donati's,  201 ;  of  1861, 202 ; 
connection  with  meteors,  211;  ele- 
ments of  periodic,  261. 

Compression,  63. 

Conjunction,  126;  inferior  and  supe- 
rior, 159. 

Constellations,  215;  list  of,  263. 

Co-ordinates,  spherical,  25. 

Corona,  96. 

Count,  least,  47. 

Crescent,  127. 

Cross-wires,  32. 

Culmination,  20. 

Cycle,  lunar,  133;  of  eclipses,  142. 

Cygni,  61,  its  distance,  222. 

Day,  solar  and  sidereal,  21 ;  inequality 
of  solar,  91;  astronomical  and  civil, 
105;  intercalary,  106. 

Declination,  20. 

Degree  of  meridian,  62. 


270 


INDEX. 


departure,  18. 

Dip  of  the  horizon,  57. 

Dipper,  215. 

Disc,  spurious,  of  stars,  223. 

Distance,  zenith,  19,'  polar,  20. 

Earth,  general  form  of,  12;  spheroidal 
form,  62;  dimensions,  63;  density, 


Gibbous,  127. 
Golden  -number,  1 34. 
Gravitation,  universal,  107. 
Heliocentric,  parallax,  56, 156;  motion 

of  planets,  157. 
Hemisphere,  17. 

Horizon,  13 ;  points  of,  19 ;  artificial,  45. 
Horizontal  point,  36. 


63;  linear  velocity  of  rotation,  70;  Hour  angle,  20;   relation   to   sidereal 

revolution  about  the  sun,  88 ;  orbital  I     time,  22. 

velocity,  89;   orbit,   89;   motion   at  Hyades,  230. 

perihelion  and  aphelion,  110;  revo-  Hyperbola,  245. 

lution  proved   by  aberration,  119; 

phases,  127 ;  elements,  257. 
Eccentricity  of  an  ellipse,  90. 
Eclipses,  135;  lunar,  135;  solar,  139; 


total,  142;  cycle  of,  142;  number  of, 
143;  of  Jupiter's  satellites,  174. 

Ecliptic,  83. 

Ecliptic  limits,  lunar,  137;  solar,  141. 

Elements,  of  planetary  orbit,  160;  of 
^ometary,  190. 

Ellipse,  244. 

Elongation,  56 ;  greatest  eastern  and 
western,  159. 

Equation,  of  centre,  102;  of  time,  104; 
annual,  133. 

Equator,  17;  celestial,  19. 

Equatorial,  40. 

Equilibrium  of  centrifugal  and  centri- 
petal forces,  107. 

Equinoctial,  19. 

Equinox,  vernal,  20,  83. 

Error  of  clock,  28. 

Errors  of  observation,  49. 

Establishment  of  port,  151. 

Evection,  132. 

Evening  star,  160, 170. 

Faculae,  96. 

Finder,  34. 

Flames,  red,  97. 

Forces,  centrifugal  and  centripetal,247. 

Foucault's  experiment,  68. 

Galaxy,  234. 
Gemini,  216. 

Geocentric,  parallax,  55;  motion  of 
planets,  155, 158,  167. 


Incidence,  angle  of,  51. 
Index  correction,  44. 


Jupiter,  173;  mass  of,  175. 

Kepler 's  laws,  111. 
Kirkwood's  law,  247. 

Latitude,  18;  equal  to  altitude  of  pole, 
23 ;  methods  of  determining,  72 ;  at 
sea,  75;  reduction  of,  75;  celestial, 
84. 

Level,  hanging,  34. 

Librations,  131. 

Light,  analysis  of,  48  ;  of  sun,  97;  velo- 
city of,  118 ;  of  planets,  183 ;  of  stars, 
218;  of  nebulse,  230. 

Line  of  sight,  32. 

Longitude,  18;  how  determined,  76;  by 
telegraph,  78;  by  star  signals,  79; 
at  sea,  80;  celestial,  84;  by  eclipses 
and  occupations,  145. 

Luculi,  96. 

Lunar  distance,  78. 

Lunation,  128. 

Magellanic  clouds,  233. 

Magnitudes,  214. 

Mars,  170. 

Mercury,  164;  transits  of,  262. 

Meridian,  18;  prime  and  celestial,  18; 

line,  19. 
Meteors,  202 ;  showers,  203;  height  and 

velocity,  205 ;  orbits,  206 ;  detonating, 

208. 
Microscope,  reading,  35. 


INDEX. 


271 


Milky  way,  234. 

Minor  planets,  171 ;  list  of,  260. 

Mira,  or  o  Ceti,  223. 

Moon,  orbit  of,  120 ;  nodes,  120  ;  obli- 
quity of  orbit,  121 ;  form  of  orbit,  122; 
line  of  apsides, 122;  meridian  zenith 
distance,  122;  distance,  123;  magni- 
tude and  mass,  125;  augmentation 
of  diameter,  125;  phases,  126;  side- 
real and  synodic  periods,  128;  retar- 
dation, 129 ;  harvest,  130 ;  rotation, 
130;  librations,  131;  other  pertur- 
bations, 132;  general  description, 
134;  elements,  257. 

Morning  star,  160, 170. 

Motion,  diurnal,  13;  upward  and 
downward,  18;  west  to  east,  88 
(note);  direct  and  retrograde,  159, 
167. 

Mural  circle,  38. 

Nadir,  18;  point,  37. 

Navigation,  sketch  of,  255. 

Nebulae,  resolvable  and  irresolvable, 
230;  annular  and  elliptic,  231  ;  spiral 
and  planetary,  232 ;  nebulous  stars, 
232;  double,  232;  variation  of  bright- 
ness, 233. 

Nebular  hypothesis,  183. 

Neptune,  182;  immense  distance  of, 
183. 

Nodes,  of  moon's  orbit,  120;  heliocentric 
longitude  of  planet's,  161. 

Noon,  105. 

Nubeculce,  233. 

Nutation,  115. 

Obliquity  of  ecliptic,  84,  115. 
Occultation,  135,  144;  of  Jupiter's  sa- 
tellites, 174. 
Octant,  48. 
Gibers' s  theory,  172. 
Opposition,  126. 
Qrion,  21 6. 


Pendulum  experiment,  68. 

Penumbra,  of  solar  spots,  95 ;  of  eolipsa 
136. 

Perigee,  122. 

Perihelion,  90. 

Perturbations,  in  earth's  orbit,  112;  in 
moon's,  130. 

Phases,  of  moon,  126 ;  of  earth,  127;  of 
Mercury  and  Venus,  165;  of  Mars, 
171. 

Photosphere,  95. 

Planetoids,  171;  list  of,  260. 

Planets,  155 ;  orbits  of,  157 ;  inferior,  158j 
stationary  points,  159,168;  elements 
of  orbit,  160;  heliocentric  longitude 
of  node,  161 ;  inclination  of  orbit, 
102;  periods,  163,168;  superior,  167  j 
distance,  169  ;  elements,  256. 

Plateau's  experiment,  185. 

Pleiades,  230. 

Points,  fixed,  36. 

Pointers,  216. 

Poles,  of  the  heavens,  15,  17;  of  the 
earth,  17. 

Pole-star,  14. 

Position  angle,  24. 

Prcesepe,  230. 

Precession,  112. 

Problem  of  three  bodies,  133. 

Projections,  spherical,  27. 

Proper  motions,  213. 

Quadrant,  48. 
Quadrature,  126. 

Radiant  points,  206. 

Rate  of  clock,  28. 

Refraction,  51;  astronomical,  52;  ge- 
neral laws,  53;  effects  of,  54. 

Resisting  medium,  196. 

Reticule,  32. 

Reirogradation,  159,  167, 

Rings  of  Saturn,  177;  disappearance 
of,  178. 


Parabola,  246. 

Parallax, geocentric  and  horizontal. 55: 

heliocentric.  5G,  156;  annual,  21<). 
Pcqaxus,  216 


os,  143. 
Satellites,  elements  of,  258. 
Saturn,  170;  rings  of,  177. 
\SeaftonSf  yo. 


272 


INDEX. 


Sextant,  42 ;  prismatic,  48. 

Shadow,  of  earth,  136;  of  moon,  139. 

Signs  of  zodiac,  84. 

Sirius,  light  of,  223  j  orbit  of,  240. 

Solar  system,  11;  orbit  of,  238. 

Solstices,  84. 

Spectroscope,  48;  use  of,  97. 

Sphere,  celestial,  11;  parallel,  16;  right 
and  oblique,  17. 

Spheroid,  oblate,  63,  245. 

Spots,  solar,  95;  observations  of,  261. 

Star  signals,  79. 

Stars,  circumpolar,  14;  fixed,  213; 
number  of,  214,234 ;  magnitudes,214 ; 
of  first  magnitude,  217 ;  constitution, 
218;  distance,  218;  differential  ob- 
servations, 220;  real  magnitudes, 
222;  variable  and  temporary,  223; 
double  and  binary, 225 ;  colored,  228; 
examples  of  variable,  266;  of  binary, 
267. 

Stationary  points,  159,  168. 

Style,  old  and  new,  106. 

Sun,  distance  of,  84;  magnitude,  87; 
rotation,  95;  constitution,  98;  irreg- 
ular advance,  102;  first  mean,  102; 
second  mean,  103;  mass  and  density, 
109;  size  compared  with  stars,  223; 
motion  in  space,  235;  elements,  256. 

Synodical  revolution,  of  moon,  128;  of 
planets,  163, 168. 

Talcotfs  method  of  finding  latitude,  74. 

Telegraph,  used  in  determining  longi- 
tude, 78. 

Telescopic  comets,  188. 

Tempel's  comet,  leads  November 
shower,  212. 

Theodolite,  48. 

Tides,  146;  daily  inequality,  148;  ge- 
neral laws,  149;  influence  of  sun, 
149;  spring  and  neap,  150;  priming 


•and  lagging,  150°;  tidal  wave,  151  j 
establishment,  151;  cotidal  lines, 
152;  height,  152;  four  daily,  153,- 
in  lakes,  154. 

Time,  solar  and  sidereal,  21 ;  sidereal 
and  right  ascension,  22 ;  Greenwich, 
77 ;  local  time  at  different  meridians, 
81;  astronomical  and  civil,  105. 

Torsion  balance,  63. 

Trade-winds,  67. 

Transit  instrument,  31. 

Transit,  20;  of  inferior  planets,  262. 

Triangle,  astronomical,  23. 

Triangulation,  60. 

Twilight,  94. 

Umbra,  of  solar  spots,  95 ;  of  eclipses, 

136. 

Universal  instrument,  48. 
Uranus,  180. 
Ursa,  major,  215;  minor,  216. 

Vanishing  points  and  circles,  26. 

Variation,  133. 

Venus,  relative  distances  from  sun  and 
earth,  84;  transit  of,  86, 165,  262;  de- 
scription of,  165. 

Vernier,  46. 

Vertical,  lines,  18;  prime,  19. 

Vulcan,  157  (note). 

Weight,  in  different  latitudes,  64;  on 
the  sun,  110. 

rear,  sidereal,  83;  tropical,  105,114; 
anomalistic,  116. 

Zenith,  18;  geographical  and  geocen- 
tric, 76. 

Zenith  telescope,  48;  use  of,  74. 
Zodiac,  84. 
Zodiacal  light,  99. 


THE   END. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


9    1962 


LD  21A-50m-12,'60 
(B6'221slO)476B 


General  Library 

University  of  California 

Berkeley 


870743 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


